Mechanical properties analysis of underground municipal pipelines subjected to traffic loads

This study delves into the mechanical properties of underground municipal pipelines under traffic loads, utilizing numerical simulations to explore the dynamic response of pipelines at varying burial locations and assess their ultimate load-bearing capacity. A simulation model was constructed using the multi-physical FEM software, focusing on the stress behavior of shallowly buried pipelines subject to traffic loading. The findings reveal that the impact of traffic loads supersedes that of soil self-weight stress for shallowly buried pipes. Pipelines located be-neath the carriageway experience greater forces compared to those beneath sidewalks, with minimal influence of centerline offset at consistent burial depths. Notably, the top, bottom, and waist portions of the pipeline are identified as particularly vulnerable regions, with rotation of the maximum force axis observed during shallow burial and deviation from the road centerline. This research provides theoretical insights for optimizing the layout of pipelines beneath urban roads, en-suring their operational safety and sustainability.

1 Introduction

The majority of municipal pipelines are directly buried within the roadbed, which play an important role in the sustainability of cities. The stress state of these pipelines within the roadbed is influenced by the traffic load acting on the road surface. Major contributors to the degradation of buried municipal pipelines include vehicular loading and roadbed compression (Tee et al., 2013; Lu et al., 2016; Rajani and Kleiner, 2001). As integral components of urban infrastructure, municipal pipelines operate within a complex subterranean environment characterized by an extensive network of buried conduits. The construction standards and quality of these pipelines do not parallel those established for structures such as buildings, roads, or tunnels. Furthermore, due to their concealed nature, identifying and rectifying damage post-occurrence presents significant challenges (Fang et al., 2018; Wang et al., 2019; Xu et al., 2021; Sulikowski and Kozubal, 2016; Yuan et al., 2025c). Damage to municipal pipelines resulting from external loads, such as traffic-induced stresses or aging, has ramifications beyond impairing their own functionality. It could potentially trigger roadbed subsidence and other associated issues. Consequently, comprehending the mechanical responses of pipelines to traffic-induced loads is of paramount significance in safeguarding against and managing potential damages (Zhang J. et al., 2016).

The issues stemming from pipeline damage have been progressively escalating over the years. Following pipeline damage, the leakage of internal fluids can lead to disasters like erosion and, in extreme cases, roadbed subsidence. Comprehensive comprehension of pipeline damage patterns and types within the soil strata assumes a pivotal role in directing pipeline reinforcement and management (Zong, 2020). Building upon the factual deficiencies found in drainage pipe systems, prevalent damage types encompass deformations, fractures, displacements, and other anomalies (Fang et al., 2019; Younis and Knight, 2010). To address these concerns, a combination of physical modeling tests, numerical simulations, and theoretical analyses were employed (Saboya et al., 2020; Rakitin and Xu, 2015; Chaallal et al., 2015b). Physical model tests offer the advantage of visually depicting the behavior of diverse pipeline parameters. Common methodologies encompass direct loading of pipeline cross-sections using indoor testing apparatus, conducting comprehensive load tests by burying pipelines in soil, and other techniques. Lay et al. applied different overlying static loads to a reinforced concrete pipeline buried in the soil layer to simulate the effects of different vehicle loads on the pipeline and observed tension cracks (Lay and Brachman, 2012). Trautmann et al. considered the effects of burial depth, soil density, pipe diameter, and pipe roughness on testing the forces on the pipe under different loads (Trautmann and Rourke, 1985). Numerical simulations broaden the scope of investigation and enable observations of conditions not feasible in experimental monitoring (Fang et al., 2018; Zhang and Huang, 2012). Illustratively, finite element software like Abaqus can be harnessed to validate the impacts of variables including pipe diameter, burial depth, load magnitude and position, foundation soil deformability, and Poisson’s ratio on the mechanical attributes of the pipe under vertical loading conditions. This approach yields outcomes akin to those derived from field tests and broadens the scope of study conditions (Zhang et al., 2022). Confronted with intricate multi-field coupling predicaments, numerical simulations circumvent the need for intricate experimental setups by systematically quantifying the influence of individual factors. This allows for meticulous sensitivity analyses to be executed (Li et al., 2019). The significance of traffic load within the aforementioned study constitutes a pivotal factor in the investigation of pipeline damage. Traffic load is a dynamic force that evolves over time, and its distribution is both non-uniform and time-varying (Alzabeebee et al., 2018). While each vehicle follows a distinct trajectory, traffic load induces repetitive impact on the road surface within a given region (Noor and Dhar, 2003; Yuan B. et al., 2025). Traffic load does not exhibit a linear progression, to delineate the attributes and patterns of traffic load, a vehicle loading model is typically formulated based on vehicle type, weight, speed, driving path, and related parameters, diverse scenarios of application might necessitate distinct models and principles (Li et al., 2017). Across numerous studies, traffic loads are frequently approximated through combinations of static loads or functions following specific patterns (Zhang et al., 2017). Cebon et al. considered these loads akin to point source vibrational forces, while Abu-Hilal deemed them stochastic mobile loads (Costanzi and Cebon, 2007; Abu-Hilal, 2000). Yu et al., on the other hand, explored subgrade sidewalk dynamics via half-sine loading dynamics (Yunyan et al., 2023). The majority of investigations employ time-varying loads featuring certain patterns to represent traffic loads. Contemporary investigations concerning buried pipelines subjected to traffic loads predominantly center on the longitudinal deformation characteristics and the cross-sectional force-deformation responses of individual conduits. For examining longitudinal deformations, the elastic foundation beam approach can be employed to elucidate the influence of external loads on the deformation, shear, and bending moment experienced by buried drainage pipes. Moreover, this methodology can account for the dynamic ramifications imposed by traffic loads (Zhang L. et al., 2016; Trickey and Moore, 2007). The primary consequence of the longitudinal force distribution is the potential for damage or complete structural failure at the junctions of buried pipelines (Chaallal et al., 2015a; Xu et al., 2017). Traffic loads are concentrated at specific cross-sections, allowing us to investigate pipe deformation from a sectional vantage point. Illustrated by a basic static load scenario, ground overloading significantly affects buried pipelines, where the apex of Von Mises stress emerges just beneath the surface loading region atop the pipe (Zhang L. et al., 2016; Yuan et al., 2025b). The depth at which the pipe is buried exhibits substantial influence on the enduring deformation of the road surface under cyclic loading conditions. Elevating the pipe burial depth corresponds to a reduction in the permanent deformation sustained by the road surface (Cao et al., 2016). Regarding the layout of individual pipelines subjected to traffic loads, research indicates that augmenting the burial depth contributes to diminishing pipeline deflection, elevating soil surface subsidence, and mitigating pipeline pressure (Ahdyeh and Hamid, 2017). Furthermore, alterations in pipe specifications, like enhancing pipe diameter and wall thickness, possess the potential to curtail the pressure induced by pipe traffic loads (Zhang et al., 2022). The circular cross-section serves as the primary foundation of the pipe’s load-bearing capability, with the cross-section primarily undergoing elliptical distortion under load (Liang et al., 2019). Ring buckling theory is a prevalent tool for characterizing the instability of materials featuring cyclic cross-sections. Solving the governing equation for buckling provides an analytical solution for the critical load in structural buckling scenarios. Deng et al. solved the differential equation for ring buckling to obtain the critical load for a non-ideal cyclic ring under homogeneous hydrostatic pressure (Deng, 2011). Research on the force characteristics of traffic loads acting on buried pipelines often solely examines the force distribution of individual pipelines within specific fixed spatial coordinates. However, owing to constraints in construction management and site environment, the positioning of buried pipelines becomes intricate and diverse. Furthermore, the intricacies arising from various departmental pipelines and the interplay among multiple pipelines need to be considered, in conjunction with the spatial positioning implications. Given its concealed nature, municipal pipelines pose challenges in repair and incur substantial costs. Anticipating its optimal burial placement and the resultant mechanical response attributes could facilitate proactive damage prevention and timely damage assessment.

This research endeavors to elucidate the force behavior of pipelines subjected to traffic loads and detect their vulnerable regions through the construction of an urban road section stratigraphic model. The investigation takes into account pipeline quantity and spatial arrangement. This endeavor not only bolsters design reliability and operational safety but also furnishes theoretical insights for subsequent pipeline reinforcement.

2 Methods

2.1 Roadbed stratigraphic model

The sidewalk and roadbed structures were determined according to the “Design Code for Urban Road Engineering” (CJJ37-2012), with a common two-way/four-lane road in the city as the standard. The cross-section included a sidewalk with wide of 15 m and a 3 m sidewalk on each side, and the entire numerical model size (width × height) was 21 m × 10 m, as shown in Figure 1. The sidewalk structure included the following: a surface layer (150 mm), a base layer (320 mm), and a sub-base layer (200 mm), which all using linear elastic material models. The value of boundary range is based on the construction methods of shallow-buried tunneling and mining in tunnel engineering. A range of five times the tunnel diameter is widely used to determine the computational domain boundary to ensure the rationality of boundary conditions (such as zero displacement). The International Tunnelling Association (ITA) and many national design codes (such as the “Highway Tunnel Design Code” of China) all recommend that the surrounding rock range be taken as three to five times the diameter of the tunnel chamber, suitable for complex geological conditions or scenarios with high-precision requirements. The roadbed structure was a homogeneous soil layer, using the soil plasticity model provided by FEM software, which yield criterion used the DP criterion to match the Mohr-Coulomb plasticity model, which ensures numerical stability and facilitates comparison with previous studies on buried pipelines. However, DP model does not incorporate strain hardening or softening behavior, nor does it capture the cumulative plastic deformation that may develop under long-term cyclic traffic loading, the present simulations focus on short-term responses to traffic loads, and the long-term accumulation of deformation and damage could be underestimated. The results of this study should mainly be interpreted as short-term (10 cycle orders) dynamic responses and local plasticization scenarios, for cumulative settlement, fatigue crack initiation and propagation, or bulging/softening effects caused by long-term cyclic loading (thousands of cycles), more advanced constitutive models or long-term test data are required for evaluation. The parameters for each material are shown in Table 1.

Figure 1

Figure 1. Road structure (mm).

Table 1

Table 1. Mechanical parameters of materials.

2.2 Pipeline modeling and location

According to the “Concrete and Reinforced Concrete Drainage Pipe” (GB/T 11,836–2009), select the appropriate model for the drainage pipe. Drainage pipe 1 has an inner diameter of 1.0 m, a wall thickness of 0.11 m, and an effective length of 2.0 m. The pipe material is C30 concrete with tensile and compressive ultimate strengths of 20 MPa and 30 MPa, respectively. The self-contained concrete model is used in the software, and the Bresler-Piser criterion is applied.

Pipe one is buried in the roadbed, with a burial depth of H m from the center of the pipe to the road surface and a distance of L m from the road centerline. The reference point when multiple pipes are buried is the center of the leftmost pipe. To investigate the influence of different pipeline arrangement combinations on the pipeline itself and ground forces, we set the burial depth H to 1m, 3 m, and 5 m; and the deviation from the center line distance L to 0 m, 3.75 m, and 9.0 m. The working conditions considered include the center line of the road, the center line of a unidirectional lane, and the center line of a sidewalk.

To investigate the mutual influence of multiple pipelines, we set up another pipeline Pipe two on the right side of Pipe one and observe its impact. The two pipelines have a horizontal distance of x meters and a vertical distance of y m. The depth of both pipelines is H = 3 m. The two pipelines are symmetrically distributed relative to the center line of the road and are at the same level height. The vertical distance y is set to 0 m, and the horizontal distance x is set to 2 m, 7.5 m and 18 m. When the two pipelines are not symmetrically distributed relative to the center line of the road, the horizontal distance x is set to 2 m, the vertical distance y is set to 0 m, and the center of the left pipeline deviates from the center line by a distance of L = 0 m, 4.75 m, and 8.5 m. The working condition designs are shown in Table 2.

Table 2

Table 2. Pipeline layout position.

2.3 Boundary condition

The current road design specifications use static load combination to describe the role of traffic load on the road surface. For a road section, vehicles impose a certain load on the road surface, which is transferred to the subgrade according to the rut, and affects the municipal pipeline, the load can also be simplified as a uniformly distributed load (Hu, 2017; Yunyan et al., 2023; Zhang et al., 2017). To represent the characteristics of vehicle motion from far and near until departure, a semi-sinusoidal curve can be used to describe the dynamic stress in the foundation soil under traffic load, as shown in Figure 2. A model with a frequency of 20 Hz, a peak load of P_max = 25 kN, and a half-sinusoidal homogeneous distribution of load is used to simulate the loading conditions on the road, as shown in Equation 1. The model is calculated for 10 cycles with a total time of 0.5 s. The traffic load was represented by a half-sine wave with fixed frequency and peak amplitude which was adopted primarily for computational efficiency and comparability with previous numerical studies on buried pipelines, neglects the effects of wheel movement, axle spacing, and multi-axle vehicle configurations, which may lead to higher stress concentrations and different temporal stress distributions. The actual vehicle loads typically exhibit mobile point/line load characteristics and contain pulse components. These factors can cause stronger local stress concentration and higher instantaneous peak values, with more complex stress paths, which may lead to more severe local damage or fatigue accumulation in the pipe body compared to this simplified model. Therefore, in the discussion, we emphasize the conclusion that the results should be limited to the simplified short-term conditions scenario, and suggest that subsequent studies should adopt moving pulse loads, actual wheel track distribution, multi-axle vehicle load spectra, or random traffic flow simulations (such as based on wheel load trains or multi-point pulse models) for sensitivity analysis to evaluate the impact of local peak values, stress spectra, and cumulative damage. The bottom of the model is a fixed constraint, while the left and right sides are roll support constraints. The sidewalk is a free boundary.

$P=P_{\mathit{max}}\times \left|\sin \left(\omega t\right)\right|(1)$

Where P is the load at a certain moment, P_max is the peak load; ω is the coefficient that determines the period, here taken as 20π, t is the action time.

Figure 2

Figure 2. Traffic load model.

3 Results and discussion

3.1 Effects of loading on pipelines

1. Pipe Stress Response during Acting Time

Calculate 0.5 s (10 cycles) for each set of conditions to obtain the force on the stratum and the drainage pipe. Taking Test T7 as an example, the force on the stratum after burying the pipe is shown in Figure 3. The cloud diagram before and after the calculation indicates that there is not much difference in the pressure of the soil layer throughout the cross-section, and the difference in the pipe cross-section is relatively large. To understand the pipeline force, a point is taken on the top (UF), bottom (DF), left side (LF), and right side (RF) of the pipeline exterior to monitor its surface pressure.

Figure 3

Figure 3. The pressure field at 0 s and 0.5 s (a) 0 s; (b) 0.5 s.

Figure 4 shows that the pressure at each location of the pipeline body undergoes significant changes with the normal force applied by the traffic load, and there are irrecoverable fluctuations with half-sine fluctuations in the load. The roadway load at the beginning (0 s) and end (0.5 s) of loading are both 25 kN/m, but there is a difference in the pipe surface force, which is caused by plastic deformation of the roadbed due to traffic load. The interaction between the pipeline and soil structure is the fundamental source of pipeline load, and elastic deformation of the roadbed structure occurs under the action of a smaller roadway load. The deformation returns to the initial state after stopping the application of the load, but a larger roadway load may. However, a larger roadway load can lead to plastic deformation of the roadbed, causing deformation and displacement of the pipeline buried in the soil layer.

Figure 4

Figure 4. Pressure of different parts of pipeline under Test T7.

Taking Test T7 as an example, the pressure zone on the inner wall of the pipe waist is enlarged at 0.5 s compared to 0 s, which indicates that the force of the pipe under the traffic load is changed, and the probability and form of damage will also be changed accordingly. The final pressures at each point are UF = 0.63 (initial value 0.59) MPa, DF = 0.67 (initial value 0.63) MPa, LF = −0.33 (initial value −0.32) MPa, and RF = −0.33 (initial value −0.32) MPa.

2. Displacement of monitoring points during the action time period

Taking T7 as an example, the displacement field in the gravity direction of the ground layer after burying the pipeline relative to the ground layer without burying the pipeline is shown in Figure 5. The cloud diagrams before and after the computation show that the gravity-directed displacement of the soil layer in the whole cross-section decreases from the ground surface to the depth, and that the pipeline has a hindering effect on the settlement in the gravity direction. To understand the displacement of the pipeline, points were taken at the top (UD), bottom (DD), left (LD), and right (RD) sides of the pipeline outside, 5.15 mm from the pipe centerline, and at the centerline of the road (RMD) and at the right shoulder of the road (RRD) to monitor the displacement of each point.

Figure 5

Figure 5. The displacement field in gravity direction at 0 s and 0.5 s (a) 0 s; (b) 0.5 s.

Figure 6 shows that the fluctuating effect of traffic loads leads to different sizes of fluctuating displacements at various points of the pipeline and the road. Similar to the force changes of the ground pipe due to traffic loading, the resulting displacements are also irreversible. For example, in case T7, the final displacements at each point are UD = 5.61 mm, DD = 5.28 mm, LD = 5.45 mm, RD = 4.55 mm, RMD = 7.68 mm, and RRD = 7.81 mm, and there are differences in deformations according to the locations of the monitoring points.

Figure 6

Figure 6. Displacement of different parts of pipeline and road surface in T7 working condition.

3.2 Mechanical response of pipes at different spatial locations

1. Single pipe force

The pressure and displacement data at the point where the peak load was located in the last cycle (t = 0.475 s) were selected for comparison. As shown in Figure 7, there are differences in the stresses on the outer surface of a single pipe with respect to its spatial location. When the distance from the centerline is the same, the upper and lower parts of the pipe are under pressure, and the deeper the depth of the pipe, the greater the pressure on the upper and lower parts of the pipe.

Figure 7

Figure 7. The force of each point at different depths under the same offset distance of a single pipeline. (a) Off-center distance L = 0 m; (b) Off-center distance L = 3.75 m; (c) Off-center distance L = 9.0 m.

Pipes buried beneath the road surface with traffic loads, the effect of depth on the pressure on the upper and lower parts of the pipe is relatively larger than that of the pipe buried beneath the sidewalk. As shown in Figure 7, for the Test T1, 4, 2, five buried beneath the sidewalk, the difference in pressure between the T1, 4, and seven groups is about 0.21 MPa, 0.25 MPa, and the difference in pressure between the T2, 5, and eight groups is about 0.20 MPa, 0.26 MPa; for the conditions buried beneath the sidewalk without direct traffic load influence in the vertical direction, the difference in pressure between the T3, 6, and nine groups is about 0.24 MPa, 0.23 MPa.

The pipe-side pressures are all in the form of negative tensile values and increase with the depth of burial. As shown in Figure 8, under the same offset distance, the tensile force on the pipe side of T1, 4, and seven is 0.18 MPa, 0.31 MPa, and 0.31 MPa, the tensile force on the pipe side of T2, 5, and eight is 0.19 MPa, 0.30 MPa, and 0.32 MPa, and the tensile force on the pipe side of T3, 6, and nine is 0.06 MPa, 0.19 MPa, and 0.31 MPa.

Figure 8

Figure 8. The force of each point of a single pipeline at different offset distances at the same depth. (a) Depth of burial H = 1 m; (b) Depth of burial H = 3 m; (c) Depth of burial H = 5 m.

The monitoring point is located on the portion of the pipe side close to the surface, and the deformation of the pipe side is caused by the compression of the upper and lower parts of the pipe annular interface resulting in vertical compression along the diameter. When there is no soil constraint around the pipe, the upper part is compressed, and the outer wall of the pipe side is supposed to be in a tensile state. Buried at a depth of 3–5 m, in the dynamic load action of the carriageway, the tension difference is not large, but with increasing depth and increasing tension on the sidewalk, it indicates that traffic load has a greater tensile influence on its buried pipeline than depth does.

When the burial depth is the same, the traffic load effect on the pipe under the sidewalk is similar in size, and the effect on the pipe under the sidewalk is relatively smaller than that under the carriageway. As shown in Figure 7, under the same burial depth, the force value of Test T1, 2, T4, 5, T7, eight are basically the same, and the pressure on the top and bottom of the pipe increases with the increase in depth. When the burial depth is shallow (1m, 3 m), the tension in waist of the pipe under the carriageway is larger than that of the pipe under the sidewalk, and when the burial depth is deeper (5 m), the size of the pipe side tension is basically the same, and the influence of traffic load is small.

2. Double piping force

As shown in Figure 9, when the two pipes are arranged symmetrically, the force of Pipe one and Pipe two is basically the same. The pressure at the top and bottom of the pipe is 0.52–0.55 MPa, and the tension at the side of the pipe is 0.28–0.31 MPa. Although the spacing between the two pipes is different, the overall difference in force is relatively small. The force of the pipe buried under the sidewalk is smaller than that under the carriageway. The pressure at the top and bottom of the pipe is 0.38–0.42 MPa, and the tension at the side of the pipe is about 0.18 MPa.

Figure 9

Figure 9. Forces at each point of the two pipes in symmetric arrangement. (a) T10 condition L = 1 m; (b) T11 condition L = 3.75 m; and (c) T12 condition L = 9.0 m.

As shown in Figure 10, in the asymmetric arrangement of double pipelines, the force of Pipeline one and Pipeline two under the carriageway is basically the same. Under carriageway traffic load conditions (T13, 14), the pressure at the top and bottom of the pipe is 0.53–0.57 MPa, and the tension at the side of the pipe is 0.29–0.30 MPa; under partially buried sidewalk conditions (T15), the force of the pipe is less than that under the carriageway, and the pressure at the top and bottom of the pipe is 0.37–0.49 MPa, with a side tensile tension of about 0.18 MPa. The burial depth H of T13-T15 groups is 3 m. It can be clearly found that, like buried pipes under the sidewalk, there is little difference in the force on the top, bottom, and side of these four pipes. At this time, the change in horizontal position has less influence on it, and the tension on the left and right sides of the pipe is basically the same. Under partially buried sidewalk conditions, the magnitude of force in each place is lower than that of buried pipelines under the carriageway.

Figure 10

Figure 10. The force of each point under the asymmetric arrangement of two pipes. (a) Test T13; (b) Test T14; (c) Test T15.

3.3 Force analysis of pipes

As shown in Figure 12, the circular cross-section of the pipe is deformed under vertical compression caused by soil weight and traffic loads. The cross-section can be regarded as changing from a circular shape to a flat oval shape. In most cases, the top and bottom of the pipe are externally compressed and internally tensioned, while the pipe side is externally tensioned and internally compressed.

Figure 12

Figure 12. Pipeline stress diagram.

Based on the results of each group, the lateral stresses at the top and bottom of the pipe are generally greater than those at the pipe waist, indicating that the longitudinal influence of traffic loads on the top and bottom of the pipe is greater relative to the lateral influence on the pipe side. For the vertical direction, the stresses at the bottom of the pipe are greater than those at the top of the pipe.

The initial damage of the pipe may occur at the top or bottom inside of the pipe, causing further deformation and instability of the pipe. The region of the pipe body with the greatest stress is basically distributed near the line of the maximum compressive stress point (peak stress axis) at the upper and lower parts of the pipe, and perpendicular to the pipe side around the line of the maximum tensile point.

To investigate the stress values of the inner and outer walls of the pipe and determine the most dangerous point, we selected the most representative conditions T1, 4 and 7 as the objects and studied their internal and external characteristics at each stress point. The stress data for each point are presented in Table 3 and Figure 11 (which the positive value indicates compression, while a negative value indicates tension). The tensile stress on the inner wall of the pipeline waist is closest to the corresponding destructive strength, making it easier to be the first to experience damage. The Mises equivalent stress can be calculated based on the shape of the mechanics of the stress shape change ratio according to energy theory, which takes into account the first, second, and third principal stresses. The influence of burial depth on pipeline response can be explained by the combined effects of traffic load transmission and soil confinement. When pipelines are shallowly buried, traffic loads are transmitted directly through the thin soil cover, resulting in significant stress concentrations at the pipe crown and invert. In this case, the dynamic response is dominated by surface loading, and stress rotation or redistribution is more pronounced with variations in offset distance. As burial depth increases, the overlying soil layer attenuates the transmitted traffic loads and provides stronger lateral confinement, leading to greater compressive stresses at the pipe waist and a larger imbalance between inner and outer wall stresses.

Table 3

Table 3. Internal and external force of pipeline under T1, 4, seven condition.

Figure 11

Figure 11. Stress and failure location of pipeline.

The Mises equivalent stress is calculated based on the specific energy theory of material mechanics regarding shape change, which takes into account the first, second, and third principal stresses. It is often used for the evaluation of fatigue, damage, and so on. For conditions T1, four and 7, the maximum Mises stress is located on the inner side of the pipe waist. The maximum Mises stresses for burial depths of 1m, 3m, and 5 m are 1.63 MPa, 2.64 MPa, and 3.65 MPa, respectively. The maximum values increase with the increase of burial depth. The ratio of the Mises stress on the inner side of the pipe waist to that on the outer side is 3.33 for condition T1, 3.34 for condition T4, and 3.54 for condition T7. It indicates that in this range, with the increase of burial depth, the imbalance of stresses between the inner and outer sides of the pipe waist increases, making it more susceptible to damage.

As shown in Figure 12, the peak stress axis of T1-T3 conditions rotates significantly with the increase of the buried pipe’s offset distance from the road center axis, and the degree of rotation decreases in T4-T6 conditions. The rotation is not obvious in T6-T9 conditions. When the burial depth is shallow, the peak stress point at the top and bottom of the pipe rotates with the increase of the offset distance from the center line, and the rotation is not obvious when the burial depth is deeper. This indicates that traffic load has a significant influence on the force and deformation of the pipe when it is shallowly buried. The offset distance between the pipe and the loaded area can change the location of its damage area. As the burial depth increases, the pipeline’s force is affected by the thickness of the overlying soil layer, and the influence of traffic load on it is weakened.

The phenomenon that the effect of load on the pipeline decreases after the depth of burial is increased may be influenced by the attenuation characteristics of traffic loads in the soil. Vehicles passing through impose loads on the ground, and these loads are transferred to the pipeline underground through the soil, and the soil is a complex material, and its physical properties and structure cause attenuation of traffic loads in the transfer process, and different soils have different dynamics, resulting in differences in the energy loss of stress in the soil. The mode of load transfer and the degree of attenuation in the soil are controlled by the type of soil, stress state, physical properties, and kinetic properties of the soil, among other factors. This is a more complex process, and this manusript does not discuss its effect in detail for the time being.

Pipeline under the action of the load generally undergoes flexural deformation, and its annular cross-section experiences tension, compression, bending, shear, torsion, and many other different actions. When a certain stress form reaches its yield limit, the pipeline may fracture at the top, bottom, and waist of the pipeline, resulting in a non-circular structure, and the sections can be regarded as articulated members after damage, resulting in differences in the longitudinal and transverse stiffnesses of the pipeline body. The pipe cross-section before damage can still be analyzed as an elastic-plastic whole, and understanding its ultimate bearing capacity after burial in the soil layer is a guide for pipeline design and maintenance. After the pipeline is pressurized, its circular cross-section flexes into a shape similar to a flat ellipse, resulting in irrecoverable plastic deformation. At this time, the functionality and safety of the pipeline are reduced. In order to evaluate the bearing capacity of the pipeline after deformation, critical loads for working conditions T1, T4 and T7 are calculated using the non-ideal circular buckling model. In addition to traffic loading, buried municipal pipelines are subjected to complex service environments that can significantly influence their mechanical performance. Factors such as groundwater level fluctuations, seasonal temperature variations, long-term subgrade settlement, and chemical corrosion jointly contribute to pipeline deterioration. These multi-field coupled effects may accelerate fatigue damage, reduce structural capacity, and shorten the service life of pipelines.

Deng (2011) used the midplane line of the ring to calculate its out-of-roundness β (Equations 2, 3) and used it as a basis to give the critical load factor λ_cr (Equation 4) in symmetric buckling, as Figure 13.

$\beta =\frac{D_{\max }-D_{\min }}{2R_0}(2)$

$R_0=\frac{D_{\max }+D_{\min }}{4}+\frac{t}{2}(3)$

${\lambda }_{cr}=3+1.153\left|\beta \right|-36.094{\beta }^2+54.201{\left|\beta \right|}^3(4)$

Figure 13

Figure 13. Non-ideal ring buckling model (Deng, 2011).

Where D_max is the maximum value of the measured outside diameter of the same cross-section of the pipe, D_min is the minimum value of the measured outside diameter of the same cross-section of the pipe, t is the thickness of the pipe wall, λ_cr is the critical load coefficient in symmetric buckling, β is the out-of-roundness of the pipe.

The critical load q_0cr of the pipe in this deformed state is obtained by bringing Equations 2–4, into Equation 5.

$q_{0cr}=\frac{{\lambda }_{cr}E}{12\left(1-{\mu }^2\right)}{\left(\frac{t}{R_0}\right)}^3(5)$

Where E is the modulus of elasticity, μ is Poisson’s ratio, and R₀ is the average radius of the center face axis of the actual ring.

The calculated parameters and results for T1, T4, and T7 conditions are presented in Table 4. The maximum and minimum diameters after deformation can be calculated based on the displacement difference between the top and bottom of the pipeline, as well as the side of the pipeline. Based on the data presented in Table 4, traffic load applied to the road surface can result in an increase in the out-of-roundness of pipeline deformation and a decrease in the critical load, indicating a deterioration in the strength of the pipeline. As the burial depth increases, the critical load of the pipeline increases, and the buried soil layer can play a certain restraining role. On one hand, it weakens the influence of traffic loads on the pipeline; on the other hand, it restricts its own deformation, which positively affects the improvement of the critical load. The calculated critical load values presented in Table 4 exhibit only slight numerical differences across different burial conditions. This outcome is primarily attributed to the insensitivity of the analytical buckling formula to small parameter variations under the chosen material and geometric conditions. The theoretical model assumes idealized ring behavior and therefore provides limited resolution for detecting minor stress or deformation differences. Consequently, the critical load values obtained here should be interpreted as indicative of overall trends—such as the increase in critical load with burial depth—rather than as precise quantitative predictions. For detailed assessment of pipeline safety margins, direct numerical simulation results, incorporating soil restraint and geometric imperfections, offer a more representative evaluation. The critical load is not sensitive to the complete circular pipe but is crucial for the defective circular pipe, which is of great significance for the quantitative assessment and usability evaluation of the increasingly numerous damaged pipelines in current cities.

Table 4

Table 4. Critical load results of working conditions T1,4,7.

To further validate the analytical buckling model, the ellipticalization of the pipeline in the numerical simulation was compared with the theoretical critical load curve. The simulated ellipticalization ratio under traffic loads showed a value slightly higher than that predicted by the theoretical formula. This difference can be attributed to the constraint effect of the surrounding soil, the nonlinear stress-strain response, and the influence of initial geometric defects, which are not considered in the idealized buckling model. The overall trend of increasing ellipticalization degree with the increase of load and burial depth is consistent between the simulation results and the theoretical results. This consistency supports the applicability of this buckling model in identifying the critical load trend, and also indicates that numerical simulations can provide more realistic assessment results when considering soil-structure interaction and material nonlinearity.

4 Conclusion

This manuscript analyzes the buried pipeline environment of municipal pipelines, explores the pipeline construction process and structure, considers various spatial burial locations of pipelines, and numerically calculates the force characteristics of buried municipal pipelines in strata under traffic loads. Based on the analysis, the following conclusions are drawn:

1. Traffic loads have a significant negative impact on pipeline safety in service, and the effect of traffic loads is greater than the effect of soil self-weight stress on the pipeline when the burial depth is shallower.

2. The location of the pipeline affects its dynamic response. The force on the pipeline buried under the traffic-loaded carriageway is greater than that under the sidewalk, and the effect of offsetting the centerline distance on the force on the pipeline is small at the same depth of burial.

3. The top, bottom, and waist sections of the pipeline are the most vulnerable areas, and buried position offset can cause rotation of the line between the top and bottom of the maximum force points. This phenomenon is more pronounced at shallower burial depths.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

YL: Writing – original draft, Writing – review and editing, Conceptualization, Data curation. LF: Methodology, Writing – review and editing. YZ: Data curation, Writing – review and editing. ZC: Resources, Validation, Visualization, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This research was funded by the Zhejiang Province General Scientific Research Project: (Y202354105). And the APC was funded by the Zhejiang Province General Scientific Research Project: (Y202354105).

Acknowledgements

The authors are grateful to the reviewers for their helpful comments on the manuscript.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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