The pipe health monitoring model with stress–strain and pipe–soil interaction for goaf Open Access

A

soil block unit above the goaf

B

soil block unit overlying A

$C$

penalty factor

$C$

damping matrix

$C1$

global optimal learning rate

$C2$

optimal learning rate of each particle iteration step

$c$

soil cohesive force

$dε$

total strain increment

$dεel$

elastic strain increment

$dεpl$

plastic strain increment

$E0$

initial elastic modulus

$erf$

probability integral function

$F$

load matrix

$f(x)$

probability density function

$G$

pipe surface parameter

$gbesti$

step-i iteration step parameter

$gbesti+1$

step-i+1 iteration step parameter

$h$

Z-direction settlement height

$K$

stiffness matrix

$K1,K2,K3,K4$

different types of kernel functions

$M$

mass matrix

$N$

contact pressure

$N′$

compressive stress of the vertical contact surface

$nr$

constitutive model parameters

$O$

soil particle structure

P(A)

probability of the lower soil block falling into the lower space

P(B)

probability of the local block falling

P(B|A)

probability of the local block sinking when the lower soil block falls

$p$

bonding stress

$q$

equivalent stress

$q′$

equivalent compressive stress

$s$

contact area

$v$

kinetic inertia

$We$

subsidence surfaces above each minimal unit quantity of soil

$W(x)$

soil settlement curve function

$W0$

the maximum subsidence

$w$

weight

$ε$

strain

$ε′$

tolerance

$ε1$⁠, $ε2$

strain

$λ1,λ2,λ3,λ4$

kernel function coefficients

$μ$

friction coefficient between the pipe and the soil

$μ1$

intrinsic friction coefficient of soil mass

$ξ$

pore water pressure in the soil

$ξi*$

loss function for the lower boundary

$ξi$

loss function for the upper boundary

$σn$

yield strength of pipeline steel

$σ1$⁠, $σ2$

stress

$ϕ$

internal friction angle of soil mass

$ω$

soil and water content

Fuel oil and natural gas are one of the main energy sources for industry, agriculture and life. Because of the uneven distribution of oil and gas resources, a reasonable resource scheduling scheme has been one of the core issue of energy development and application. Pipeline is widely used in long distance transportation of oil and gas and often arranged in the region which is difficult to manually manage. Thus, study the design method and the health status monitor model is crucial to ensure the safety of oil and gas transportation process. This paper mainly focused on the phenomenon of the pipeline deformation, crack and fracture in goaf which is the soil subsidence area caused by the mining and tunnel excavation. The pipeline will deform, crack or even break under the dual action of self-weight and soil extrusion in goaf. Numerous scholars both have studied the external loading situation, dynamic modelling, design method and health status monitoring techniques of pipelines in goaf as follows:

The external load on buried pipeline passing through the mining areas primarily arises from soil subsidence displacement and gravity. For the study of soil subsidence displacement, the theory of stochastic medium was pioneered by Polish scholar Litwiniszyn, positing that defining the elasticity, plasticity, continuity, and looseness of geotechnical medium in terms of continuity and discontinuity is challenging, thus generalising the geotechnical medium as stochastic mediums (Litwiniszyn, 1974). Building upon this work, Liu Baochen further idealised the model of the medium to establish the probabilistic integral method (Baochen, 1993), which has become the most widely utilised technique for predicting the subsidence displacement of the soil body. Alvarez-Fernandez et al. considered the influence of floor strata movement in steep coal seam mining, conducted an in-depth study on the calculation of surface movement, and obtained the influence function n–k of rock strata movement (Alvarez-fernandez et al., 2005; Gonzalez-nicieza et al., 2005).

Methods for investigating the influence of surface subsidence on pipeline health encompass both analysis algorithm and numerical simulation approaches. In analytical research, Sarvanis and Karamanos (2017) established a pipeline strain prediction model, providing a basis for preliminary pipeline design. Cheuk et al. (2007a, 2007b) conducted an experimental study on the vertical and horizontal resistance of buried pipelines in soft clay, and obtained the relationship curves between the vertical and horizontal displacements of buried pipelines and corresponding resistance. Hindy and Novak (1979) proposed the simulation method of pipeline stress considering the influence of earthquake. Furthermore, Wang and Yao (2008) and Huang et al. (2019) established correlations between pipeline deflection, bending moment, stress, strain and soil subsidence using the elastic foundation beam model. Although analytical methods yield precise results, they often involve complex computations. With the developments of computer technology, the numerical simulation method has rapidly progressed. Lim et al. (2001) established the strain response of buried pipeline under different working conditions based on soil spring model. Kokavessis and Anagnostidis (2006) used finite element analysis (FEA) to simulate pipe–soil interaction with non-linear contact theory, concluding its superiority over the soil spring model. Sharif and Shajib (2023) analysed the pipe–soil interaction behaviour based on the finite element method and analysed the effect of different parameters of the pipe on its interaction. Faghih et al. (2024) established a monitoring system based on the pipeline axial stress and strain, comparing it with experimental data to enhance feedback and concluded that the pipeline axial tensile stress as an indicator pipeline health status monitoring. Yan et al. (2018) proposed a high-precision diameter detection technology for oil and gas pipelines to accurately detect pipeline deformation. By detecting the shape of the inner wall of the pipeline, analysing the attitude and position relationship between the calliper and the pipeline, a three-dimensional reconstruction model of the pipeline is established based on coordinate transformation and least square method.

Through the study of soil subsidence curve and the FEM method, the dynamic characteristics of pipe–soil interaction process of soil subsidence in goaf were deeply explored. However, all the above methods use FEA method to establish the dynamic model of pipe–soil. The FEA method has a slower solution speed, and the settlement process needs several hours to be solved, which makes it difficult to directly apply the continuous model in the health monitoring system. Therefore, on the basis of previous studies, this paper proposed a pipe stress–strain-based health monitoring model considering the interaction mechanism of pipe–soil in goaf (Figure 1). In this paper, the time–space motion law of soil subsidence process was first clarified by probability integral method. Through the pipe–soil contact conditions and the dynamics model of the pipe and soil, the pipe-soil interaction mechanism was studied. Based on the FEM method, the stress and strain of the pipe in goaf could be solved. Based on the analysis results which is verified the accuracy by the experiments, the main factors affecting pipeline strain were studied by covariance analysis. And the mixed kernel functions support vector machine were proposed in this paper to establish the stress–strain-based health monitoring model and the coefficients of this model were solved by the particle swarm optimisation which can prove the precision of the pipe stress–strain-based health monitoring model. The paper design a mixed kernel function that can cover the strain feature space of the pipeline, and optimise the coverage range of the mixed kernel function based on the particle swarm optimisation method to improve the model accuracy. Through this method, the regularities of the discrete data obtained from FEA are summarised, and a pipeline health state monitoring model that can quickly calculate the mapping relationship between soil subsidence state, pipeline strain state and pipeline stress state is obtained. Because this model is just a non-differential functions whose coefficients are determined and it no need for complex interpolation models and functional solving, the model will perform more efficient than the FEM in the engineering.

Soil medium is different from elastomer, plastic media and other continuous media, and its motion law is random. To describe the space–time motion law accurately, the paper studied the probability integral method. The motion law of the medium is observed from different scales. The soil is regarded as a discrete medium firstly, and the simply motion model of the discrete medium is studied based on the probability method. The precise motion process of the continuous medium is derived by the integral method. And the soil subsidence process can be described by this model (Figure 2(a)). The coordinate system O-XZ is established, where the Z-direction is the subsidence direction and the X-direction is the influence direction of the mining unit. The law of soil movement is studied by macroscopic scale firstly. Assume that the soil mass is a whole fixed clod. Clods are indivisible. And each clod can move independently. The size of the clod is 1 × 1. 1 refers to the unit length. When soil block A is excavated under the soil mass, the soil mass begins to sink due to the influence of its own weight. In the layer above the excavated soil block A, there are two soil blocks that may fall into the lowest space. Since the two soil blocks are in the same state, the probability of falling into the lowest space is 0.5. At this time, there are four clods above the two clods, only if the clods below the four clods fall, these four clods will fall down. Therefore, the prior probability can be used to calculate the falling probability of these four clods, as shown in Equation 1. P(A) is the probability of the lower soil block falling into the lower space, P(B) is the probability of the local block falling, and P(B|A) is the probability of the local block sinking when the lower soil block falls. And the probability of all the clods falling can be calculated.

Then the study scale is reduced. When soil blocks are replaced by particles equivalent, the subsidence space can be divided into infinitesimally small spaces, and each subsidence space of soil mass can be regarded as a space with length $δ$ in the x-direction and height in the z-direction of h, which $δ$ is a very small amount. Equation 1 can also be used to calculate the subsidence probability of each space. When the study scale is reduced, the continuous space can be regarded as full of random particles (Fan et al., 2014; Luo, 2015). So the spatial subsidence probability can also be regarded as the number of particles falling in the space. Therefore, the spatial subsidence height of soil mass can be expressed as Equation 2, that is, the proportion of settling particles multiplied by the spatial height h. The soil subsidence curve is similar to the normal distribution curve (Figure 2(b)).

The soil subsidence curve is established through the multi-scale research method. Because of the isotropy of the material, the influence of the soil with length $δ$ and height h on the surface settlement can be obtained by rotating the cross-section soil subsidence curve (Figure 2(c)).

However, in general, the volume of excavated soil is large. So it is necessary to use the integral method to obtain the soil subsidence curve which can describe the influence of the actual goaf. Assume that there are n minimal unit quantities of soil (Figure 2(c)) in the excavation area at this time, which are accumulated to form a cylinder with radius r and height h. At this time, there must be subsidence surfaces just above each minimal unit quantity of soil as shown in Equation 3.

The influence of the overall excavation area is the accumulation of each subsidence surface (Figure 2(d)). Integrates Equation 3 can obtain the model as the Equation 4 shows.

The erf is the probability integral function which can be described as Equation 5. W₀ is the maximum subsidence.

According to the above method and the experimental result, the soil subsidence surface is calculated (Figure 3). Since soil subsidence process is divided into three stages: incomplete mining, critical full mining and full mining. The soil settlement and displacement curve is divided into six stages to better characterise the slow subsidence process, among which the first three stages are incomplete mining. The last three stages are full mining state.

Partial differential equations of pipe–soil is established to describe the interaction mechanism of pipe-soil in goaf in this paper. Combined with the soil subsidence surface, the stress and strain of pipe in the process of soil subsidence is solved based on the FEM. According to the analysis results, the basis of pipe health monitoring model can be established (Zhuang et al., 2021). In this paper, the model of dynamic process of pipe–soil interaction in goaf is studied, and the dynamic characteristics of pipe–soil in the process of soil settlement in goaf are solved by ABAQUS.

The subsidence process of goaf can be divided into three stages. In the first stage, the soil is elastic deformation and the pipeline is elastic deformation; in the second stage, the soil strain exceeds the yield surface and reaches the plastic flow stage. Since the yield limit of the pipeline is higher than that of the soil, the pipeline is still elastic deformation at this stage; in the third stage, the stress of the pipeline exceeds the yield limit of the pipeline due to the excessive pressure of the soil subsidence. At this time, the pipe also began to produce plastic deformation. In the three stages of soil subsidence, pipe and soil interact with each other through viscous force and friction. The interaction process between soil and pipeline in the stage of soil subsidence is shown in Figure 4. The elastic dynamic model of the pipeline and the stress–strain differential equation of the plastic deformation process are established respectively. The elastic dynamic model is shown in Equation 6. The stiffness matrix K, mass matrix M and damping matrix C in the equation can be established based on the elastic modulus E, Poisson’s ratio v, density $ρ$⁠, material damping c and pipe geometric parameters such as section geometry parameters, length and polar moment of inertia.

The elastic deformation of the pipeline is caused by the contact force between soil and pipe during the soil subsidence process. The contact force model is established considering the friction coefficient, as shown in Equation 5. The coefficient $μ$ is the friction coefficient between the pipe and the soil. The coefficient $μ1$ is the intrinsic friction coefficient of soil mass. The parameter $N$ is the contact pressure. The parameter $p$ is the bonding stress. The parameter $s$ is the contact area. The mechanical model can be simplified to the ultimate wear resistance, as shown in Equation 6. The parameter $O$ is the soil particle structure. The parameter $ω$ is the soil and water content. The parameter $ξ$ is the pore water pressure in the soil. The parameter $N′$ is the compressive stress of the vertical contact surface. The parameter $G$ is the pipe surface parameter. In the position where the soil does not sink, the constraint of the soil on the pipe can be equivalent to the spring constraint. As shown in Figure 5, the constraint of the soil pair in the fixed section on the pipeline is regarded as the elastic equivalent constraint in the X, Y and Z directions, and the equivalent stiffness of the soil spring can be calculated by Equations 7 and 8 (Pore et al., 2021).

When the pipe stress exceeds the yield limit, plastic deformation occurs, and the strain differential equation can be established. And the stress and strain no longer show a linear relationship when the plastic deformation occurs. To describe the stress–strain relationship of pipeline materials more accurately, the non-linear constitutive equation of pipe materials is proposed. The Ramberg–Osgood model is often used to simulate the plastic deformation process of pipe during the soil subsidence, as shown in Equation 9, where the parameter $E0$ is the initial elastic modulus. The parameter $σn$ is the yield strength of pipeline steel. The parameters $n$ and $r$ are constitutive model parameters (ASME, 2003).

Because the soil is a random particle medium, its plastic constitutive equation is different from that of metal pipes. The constitutive equation of elastic-plastic strain process of soil with single yield surface was summarised by previous experiments. Mohr–Columb model was introduced as the constitutive equation for soil deformation characterisation, as Equation 10 shown. The parameter $dε$ is the total strain increment. The parameter $dεel$ is the elastic strain increment; $dεpl$ is the plastic strain increment. Considering the shear stress on the soil and the contact Angle between particles, the critical conditions for buckling deformation are shown in Equation 11. The parameter $q$ is the equivalent stress. The parameter $q′$ is the equivalent compressive stress. The parameter $c$ is soil cohesive force. The parameter $ϕ$ is the internal friction angle of soil mass.

Above theoretical model is solved by FEA method in ABAQUS. First, considering the elastic modulus, Poisson’s ratio and density parameters of pipe materials, the linear dynamic equation of pipe elastic deformation stage is established based on FEA method. Based on the buckling limit and the constitutive equation of the pipe, the boundary conditions of the stress–strain differential equation in the pipe plastic deformation process are established. Then the elastoplastic deformation process of the pipeline is described by the above equations. Mohr–Coulomb model is used to describe the elastic-plastic flow in the process of soil settlement, and the yield surface or flow potential surface of soil is established considering the friction Angle of soil as the boundary condition of elastic-plastic transformation. According to Equation 10, the strain of soil in the process of elastic-plastic deformation is solved. Combining the mathematical models shown in Equations 7 and 8, the pipe deformation model related to pipe–soil contact area and friction coefficient is established, which can be solved by Newton iteration method.

To establish the stress–strain solution function of pipe and soil, the finite elements of pipe and soil should be chose firstly according to the geometric parameters and deformation trend of pipe and soil. A reasonable finite element grid layout scheme should be designed and appropriate interpolation nodes of the solution function should be established to avoid distortion of the FEA results. Solid mesh is selected for both the pipeline mesh and the soil mesh. The zoning modelling method is adopted to divide the grid division area into the region with pipe and the region without pipe. The pipe area is cut from the location of the pipe as the centre to the surrounding area, and the solid grid is divided by radiating from the centre to the edge line. In the area without pipe, the solid grid is divided evenly according to the grid length on the realisation. The grid of the pipeline and soil is respectively established (Figure 6).

The load in the pipe–soil model mainly includes the weight of the pipe and the soil, the pressure of the fluid in the pipe and the displacement of the soil. The weight of the pipe and soil can be loaded by setting the material density and gravitational acceleration of the model. The fluid pressure can be directly achieved by setting the inner wall pressure of the pipe. Since the influence range of soil subsidence in the axial direction of the pipeline is different at each stage, the soil bottom surface is symmetrically divided into 12 regions by zoning, and 6 loading steps are set to realise the soil subsidence displacement curve at each stage to be loaded in the corresponding region respectively. The regions of subsidence displacement loading at different stages (Figure 7).

The “hard contact” contact relation is set in the normal direction of the pipe–soil contact surface to represent the normal separation behaviour of the pipe and soil. The contact constraints are applied in the normal direction when there is no gap between the pipe and the surrounding soil. If the gap between the pipe and the soil around the pipe is greater than 0, the pipe and soil are considered to be separated, and the contact constraints at the nodes are canceled. When the gap between the tube and the soil around the tube is greater than 0, the tube and soil are considered to be separated and the contact constraint at the node is deactivated. The friction coefficient of pipe–soil is set at 0.5 in the tangential direction of pipe–soil contact surface, and the penalty function is used to realise the calculation of friction resistance of pipe–soil contact surface.

The three-segment polyline model is used to describe the material constitutive model. This model represents the stress–strain patterns of the pipe material in the elastic deformation phase, the plastic strengthening phase and the plastic deformation phase. The equation of the model is described as Equation 12 (Ministry of Housing and Urban-Rural Development of the People's Republic of China [MOHURD], 2017). In Equation 12, the parameter E₁ means the elastic modulus. The parameter E₂ describes the linear relationship of the strain and stresses in the plastic strengthening stage. The $σ1$ is the critical stress between the elastic deformation stage and plastic strengthening stage. And the $ε1$ is the critical strain between the elastic deformation stage and plastic strengthening stage. The $σ2$ is the critical stress between the plastic strengthening stage and the yielding stage. And the $ε2$ is the critical strain between the plastic strengthening stage and the yielding stage. The material parameters for setting pipe soil are shown in Tables 1 and 2. Through numerical simulation, the overall pipe–soil deformation cloud map obtained is shown in Figure 8. The axial stress and radial deformation cloud map of each stage of the pipeline are shown in Figures 9 and 10. The overall deformation of the pipe–soil, as shown in Figure 8, exhibits symmetry, with the maximum subsidence displacement occurring at the centre. The soil body is driven to settle from the middle towards the two ends, with the settlement displacement gradually decreases. The maximum settlement displacement of the soil body is 0.15 m.

From Figure 9, it can be seen that the axial stress of the pipe is the highest occurs at the middle position of the pipe in stages 1 and 2. The gravitational force of the soil body causes bending deformation, resulting in compressive stress on the upper surface of the pipe with a negative stress value, while tensile stress is on the lower surface with a positive stress value. The value of the tensile and compressive stress are similar to the value of the pipe due to the symmetrical structure. With increasing soil subsidence displacement, starting from stage 3, three peaks of axial stress emerge in the pipe, with the largest peak occurring at the middle, and the other two peaks are on both sides. Subsequently, in stages 4, 5 and 6, the stress peaks on both sides progressively increase, and the position in the axial direction of the pipe gradually shift outward. Figure 10 depicts a consistent parabolic trend with upward opening in the radial deformation of the pipe, with the largest deformation occurring at the middle position of the pipe.

To further analyse the distribution law of the axial strain of the pipeline, considering the influence of the internal pressure, burial depth, pipe diameter and thickness of the pipeline on the stress, a multi-factor and multi-variable parameter set with the factor variables of 0, 2, 4 and 6 MPa for the internal pressure of the pipeline, the variables of 0.2, 0.35, 0.5 and 0.6 m for the burial depth, the variables of 25, 50, 80 and 100 mm for the diameter of the pipeline, and the variables of 3.25, 4, 5 and 5.5 mm for the thickness of the pipeline, is designed. A randomised combination was used to construct 20 sets of simulation parameter groups of influencing factors as shown in Table 3.

Based on the numerical simulation method of pipe–soil settlement process according to Equation 5, combined with the simulation parameters in Table 3, its FEA model is sequentially established for the simulation of the settlement process. The axial paths at the top and bottom of the pipe are created by using the node list, and the axial stress and radial deformation curves of the pipe are output respectively as shown in Figure 11. From Figures 11(a) and 11(b), it can be observed that the trend of the axial stress values at the top and bottom of the pipe is the same. However, there is a distinction wherein the bottom of the pipe is tensile stress, while the top of the pipe at the corresponding position is compressive stress, leading to differences in the positive and negative signs. Consequently, stress analysis focuses solely on the axial stress at the bottom of the pipe. Analysis of the 20 stress curves in Figure 11(a) reveals three distinct trends. Trend 1, the stress curve has only one wave peak with maximum stress occurring at the middle position of the pipe; Trend 2, the stress curve has three wave peaks, and the wave peak at the middle position of the pipe is very close to the value of the stress around it, but it is much larger than the value of the stress at the wave peaks at the two sides of the pipe, and the position of the wave peaks at the two sides of the pipe in the axial direction is different due to the different process parameters; Trend 3, there are still three wave peaks in the stress curves, differ with the trend 2, the maximum stress value closely aligned, and even the maximum stress of the pipeline under some process parameters is not in the middle position of the pipeline. Under the different process parameters, the distribution location of the hazardous cross-section of the pipeline has great differences, so it is very important to establish the axial stress distribution model of the pipeline.

From Figure 11(c), it can be seen that the radial deformation of the pipeline presents two states. When the maximum deformation of the pipeline is small, the radial deformation curve follows an upward-opening parabolic shape, with the largest deformation occurring at the middle position. Conversely, when the maximum deformation of the pipeline is large, a more pronounced point of inflexion occurs near the middle position, resulting in the largest radial deformation at this point.

Due to the inefficiency of the FEA method in predicting pipeline stress models during subsidence in goaf areas, which makes it challenging to apply in practical condition monitoring, it is paramount to establish a high-precision and high-efficiency mapping model of subsidence parameters – pipeline geometric parameters – pipeline stress based on the deformation mechanism of pipelines during subsidence. The stress–strain curve characteristics of pipelines induced by goaf subsidence are complex, and using a support vector machine (SVM) method with a single kernel function to generalise these patterns can easily lead to feature loss, resulting in insufficient interpolation or generalisation capabilities (Wei et al., 2024). Therefore, this paper proposes a method using a hybrid kernel function support vector machine (HK-SVM), and optimises the parameters and weights of the hybrid kernel function based on a particle swarm optimisation algorithm with accuracy as the objective, to obtain an optimal model that can encompass the deformation characteristics of the pipeline.

SVM methods are commonly used in regression or classification models with small sample sizes, the main principle is illustrated in Figure 12. Given the complexity of the mapping relationships between subsidence parameters, pipeline geometric parameters and pipeline stress, the original feature space established using these parameters results in a complex surface. To simplify the representation of the model’s geometric features while summarising the stress–strain field patterns, SVM uses kernel functions to map sample points to a higher-dimensional space. This process transforms the features of the sample points, selecting a feature space where the sample points are close to a hyperplane, and utilises this hyperplane to inductively summarise the patterns.

SVM regression model is given by Equation 13. First, a hyperplane is established in the feature space, and based on this hyperplane, a margin band is created. The distance from the margin band to the hyperplane is the tolerance $ε′$ (Tong et al., 2009). When a sample point falls outside the margin band, its distance from the band is calculated and considered as the loss function. In the middle of $ξi$ and $ξi*$ introducing slack variables for the upper and lower boundaries of the margin band into the equation, the objective function is obtained as shown in Equation 14. Minimising this objective function increases the number of sample points within the tolerance band while simultaneously minimising the total distance of sample points outside the tolerance band to the tolerance band.

$ξi$ is the loss function for the upper boundary. $ξi *$ is the loss function for the lower boundary, and $ε′$ is the tolerance written in the text.

Combining the constraint equations, the Lagrangian function is established as shown in Equation 15. Due to satisfaction of the constraint conditions and the minimisation of the objective function at this stage, the dual problem can be formulated as shown in Equations 16 and 17. Given the monotonicity of the hyperplane, the maximisation term using the max operator addresses the boundary conditions to investigate the minimisation problem. Meanwhile, the min term seeks the solution where the gradient of the objective function and boundary conditions equals zero, representing the minimum value on the boundary. Solutions that do not satisfy the second condition are in a growth phase within the dual problem, and their values necessarily exceed those of the minimum points. By employing gradient descent to solve the dual problem, the spatial projection relationship satisfying the boundary conditions and the minimum value condition of the objective function can be obtained, along with its hyperplane.

The kernel function in SVM projects sample points into a new feature space, defining the basis vectors of this space. Therefore, selecting a kernel function that closely adheres to the patterns in the data is crucial for accurately constructing SVM models. Commonly used SVM kernel functions include linear, polynomial, Gaussian and radial basis function kernels. Each kernel function is suitable for different scenarios, for instance, polynomial kernels, with their lower model flexibility, are often used to capture overall patterns in problems and exhibit good generalisation capabilities. On the other hand, radial basis function kernels offer higher flexibility and interpolation capabilities. To leverage the characteristics of different kernel functions (Ebru and Gurkan, 2022), this paper constructs a hybrid kernel function, as shown in Equation 18, where K₁, K₂, K₃, K₄ are different types of kernel functions, and λ₁, λ₂, λ₃, λ₄ are kernel function coefficients.

To optimise the weights of the hybrid kernel function, this study treats kernel function parameters and weights as position vectors of particles in a particle swarm. The objective function is the prediction accuracy of the SVM, incorporating the optimisation process of hybrid kernel function parameters and weights. The principle is illustrated in Figure 13: initially, four types of kernel functions are blended, and their weights and parameters are extracted as particle positions. A total of 100 sets of particle positions, representing hybrid kernel functions, are randomly generated. These hybrid kernel functions are then used in SVM to predict the output values of various samples and compute the model accuracy. The optimisation iterates by updating the best values for each particle and the global optimum. Utilising the global best, individual bests, and the velocity vectors of particles from the previous iteration, the velocity vector for the current iteration is computed as shown in Equation 19, where v is the kinetic inertia, C₁ is the global optimal learning rate, and C₂ is the optimal learning rate of each particle iteration step. By combining the learning rate and velocity vector, the particle positions are updated iteratively until the optimal model is found. The convergence condition is given by Equation 20, where $gbesti+1$ is step-i + 1 iteration step parameter, $gbesti$ is step-i iteration step parameter.

The weights and parameters of the hybrid kernel function are shown in Table 4. The pipeline health monitoring model established based on the hybrid kernel function is depicted in Figure 14. It describe the relationship of the stresses, the position of pipe and the subsidence displacement. That we could predict the stress of the pipe based on the subsidence displacement. And the engineers can decide whether to carry out maintenance according to the result which is calculated based on this model. The training convergence process is shown in Figure 15. It means the accuracy which influenced by the parameters γ and C of hybrid kernel. And the comparison results of accuracy with single kernel function support vector machines are presented in Table 5. Mean squared error (MSE), root mean squared error (RMSE) and R square are the evaluation indicators for the accuracy of SVM model. It can be observed that the hybrid kernel function support vector machine outperforms the single kernel function SVM in terms of both generalisation and interpolation capabilities. The cross-validation loss diagram is shown as Figure 16.

To evaluate the accuracy of the model, an experimental platform as shown in Figure 17 was established. The framework of the experimental platform was built using a steel frame structure. The experimental platform was 6 m, 1 m wide and 1.475 m in working height. Within the 3-m range from the middle to both sides beneath the experimental platform, 10 pairs of jacks were evenly distributed at intervals of 0.25 m, and the initial height of the jacks was 0.5 m as shown in Figure16(c). The two sides and the back of the experimental platform were fixed using steel plates and spot welding. For the front side, to facilitate the observation of the settlement of the soil mass, acrylic plates and bolts were used to fix it on the steel structure and reinforced with wooden strips. Before the experiment, the soil with a humidity of 16% was first filled inside the experimental platform and pipes were buried, with a burial depth of 0.8 m. The parameters of the experimental pipe were: L 6.3 m, D 85 mm, t 4 mm, and the pipe material was ×80. Distributed optical fibre sensors were pasted on the top and bottom of the pipe to collect its axial strain. Then, heavy objects were placed on top of the soil mass for pressure maintaining. The overall weight of the pressure maintaining objects was 800 kg, evenly distributed on the top surface of the soil mass of the experimental platform, to make the internal pressure of the soil mass close to that of the real ground.

According to the soil settlement displacement curve shown in Figure 3, the height of the jack is adjusted in stages to simulate the settlement of the soil at each stage. The distributed optical fibre sensing system is set with a collection frequency of 10 Hz and an axial interval of 2.6 mm to measure the axial strain of the pipeline. The axial strain results of settlement stages 1, 3 and 6 obtained from the experimental measurement and the corresponding simulation results are shown in Figure 18. Among them, the error between the maximum strain and the actual strain does not exceed 6%. The overall trend of the experimental data is the same as that of the simulation data. It can be known that the numerical simulation accuracy is sufficient to well reflect the maximum strain and even the variable distribution trend during the subsidence process of the goaf. Therefore, the health status monitoring model established based on the numerical simulation results can better reflect the pipeline strain at each stage of the goaf subsidence and has great significance for the construction of the pipeline health status monitoring system.

The paper integrates probabilistic integration methods and numerical simulation techniques to study the elastoplastic deformation mechanisms of pipelines during goaf subsidence. Building upon this foundation, it investigates the relationship models between design parameters and stress outcomes. The study utilises particle swarm optimisation to optimise parameters for a hybrid kernel function support vector machine, establishing a pipeline stress–strain-based health monitoring model. Experimental validation confirms the accuracy and effectiveness of simulation and model predictions. The final modelling accuracy reaches 99.7%, demonstrating the efficacy of the model. The main contributions of the article are as follows:

1. Explored probability integration theory and applied it to characterise the subsidence process of goaf soil, elucidating the subsidence curve of goaf soil.

2. Clarified the dynamic characteristics of pipelines and soil during goaf subsidence, investigating the elastoplastic deformation of pipelines and soil elastoplastic flow processes.

3. Combined the mechanism of pipeline elastoplastic deformation to propose optimal design solutions for pipelines.

4. Introduced a particle swarm optimisation-based method for optimising a hybrid kernel function support vector machine, aiming to maximise feature coverage and accuracy in the model.