When pipeline is subjected to transverse landslide, it will produce large deformations, and the pipe-soil shall present complex nonlinear contact status. In order to simulate the corresponding behavior of pipeline, Shell181 element is took to simulate the natural gas pipe. Although soil consists of soil particles, but it often appears as entities in nature, so it can be regarded as one kind solid with elasticity and plasticity. In order to simulate the non-slip area soils more realistically, Solid185 element is used. Between the pipe and soil, the contact relationship is set up, which use TARGE170 unit and CONTA174 unit as the target surface element and contact surface element respectively.

Here, X80 pipe is used as the pipe modeling material, and the parameters are shown in Table 1. For the pipe stress-strain relationship curve, the ideal elastic-plastic bilinear kinematic hardening model is taken and shown in Figure 2. Outside the landslide zone, the soil parameters are adopted as follows: the density is 2000 kg/m³, the elastic modulus is 32.5 MPa, the poisson’s ratio is 0.35, the internal cohesion is 15 kPa, and the internal friction angle takes 10˚. It is worth noting that the weight of the natural gas is averaged distribution into the pipe body, so the natural gas pipeline equivalent density ρ_eq. is 9197.7 kg/m³.

The engineering practice shows that the soil mass in the non-slip area is basically in elastic state, so the soil can be regarded as an elastic material. The semi-symmetric finite element model (FEM model) is set up as Figure 3.

The length of the natural gas pipe is five times as long as the half-length pushed by the transverse landslide. The non-slip soil bulk width along the pipe transverse is 12 times long as the pipe diameter, but along the pipe axial direction

Notes: pipe diameter D (mm), pipe thickness δ (mm), natural gas density γ (kN/m³), internal pressure P (MPa), pipe density ρ (kg/m³), pipe material yield strength σ₁ (MPa), poisson’s ratio μ, the elastic modulus E₁ (GPa), the tangent modulus E₂ (MPa).

is 4 times long as the half-length loaded by the transverse landslide. In the soil bulk, the upper surface and one end face near the landslide is free surface, but another end face far from the landslide is constrained along the pipe axis direction. The front and back plane of non-slip soil bulk are fixed up perpendicular to the corresponding surface. The soil base is fixed in all directions. On the natural gas pipe’s two ends, one end away from landslide is constrained along the pipe axis, and the other end fixed as symmetry constraint.

The research [12] has shown that: under the action of transverse landslide, the pipe experiences different thrust force in different buried depth. Hereby, the thrust force on pipe in different buried depth is calculated and listed as follows (Table 2). About the distribution of thrust force along the pipe due to landslide, the study [20] have assumed it as uniform distribution, but in actual, the load distribution along the pipe is more closer to the parabolic distribution. Therefore, it will be assumed as a parabolic distribution in this article, and the parabolic vertex load is calculated as the study [12] .

In order to get the limit length value of the natural gas pipeline safely bearing transverse landslide, the pipeline failure criteria is on the basis of that: the pipe Von Mises stress reach the pipe material yield strength. After a large number of trial, the final results is listed as follows (Table 3), and the relevant graphs are

Notes: pipe thickness δ (mm), buried depth of pipeline H (m), angle between horizontal plane and soil sliding direction in place where pipe buried β (˚), uniform load q (kN/m).

Notes: buried depth of pipeline H (m), pipe length pushed by landslide process L (m), angle between horizontal plane and soil sliding direction in place where pipe buried β (˚), the max Von Mises stress of pipe σ_max (MPa), the max displacement of pipe U_max (m).

drawn such as (Figures 4-7). At the same time, in order to observe the stress response characteristics of pipeline under the landslide, the finite element Mises stress contour plot of pipeline (it is buried in 2 m depth; the pipe length loaded by landslide process is 27 m; the angle between horizontal plane and soil sliding direction in place where pipe buried is 30˚) is taken as an example shown in Figure 8.

Observing Figures 4-7, it is easy to draw the following conclusions:

1) With the increase in β value, the σ_max value decreases gradually when the same L value is adopted.

2) With the increase in H value, the σ_max value increases gradually when the same L value is adopted.

3) With the increase of H value, the limit length of pipeline safely withstanding transverse landslide reduces gradually, and the curve presents a good linearity.

Figure 8 shows that: under the landslide thrust force, Mises stress in middle part of the pipeline and interface of soil slip zone and non-slip zone, is larger than the other positions. Such phenomena is in accordance with the practice discovery: the pipeline cracks or breaks from the middle part of pipeline and interface of soil slip zone and non-slip zone. Of course, this also indirectly indicates the feasibility of the above modeling.

To test and verify the reliability of above results, slope and pipe three-dimensional finite element model is set up using the ABAQUS finite element software (Figure 9).

The back of this Model is 70 m high; the front is 25 m high; the bottom is 100 m long, 100 m wide; the length of pipeline is 120 m. The bottom of the bedrock is fixed; two end faces, front and back end faces are constrained along the vertical direction of these end faces; the pipeline is embedded in soil. The stress-strain relationship of pipeline is the same as Figure 2. The buried depth of pipeline is 2 m, other parameters refer to Table 1. Mohr-Coulomb model is used to express the constitutive relationship of upper soil. The bedrock possessing 15 m thickness is regarded as an elastomer material. The bedrock and soil parameters all are shown in Table 4.

Notes: the elastic modulus E (MPa), the poisson’s ratio μ, density γ (kN/m³), internal friction angle φ (˚), cohesion c (kPa).

The gravity and pipeline internal pressure is applied to the model, then the Strength Reduction Method [21] is utilized to the 27 m width zone in the model central area until the stability of slope reach to the state of limit equilibrium. With partial soil turning into plastic state, the failure surfaces under limit shear situation is produced gradually (Figure 10).

When the strength reduction factor surpasses 1.87, the slope is in unstable state, and it will change into landslide. In this state, the displacement of the slip soil is shown as Figure 11, and the pipe Mises stress contour plot is shown in Figure 12.

The Figure 11 shows that the maximal displacement of slope can be up to 0.5768 m. Meanwhile, the Figure 12 shows the maximal displacement of pipeline is 0.4937 m, and the maximum Von Mises stress of pipeline reach to 530 MPa. While landslide happens, the maximum stress and displacement of pipeline are greater than the ones above. The initial internal friction angle φ and cohesion c of soil are reduced to 14˚ and 10.695 kPa when the strength reduction factor reaches to 1.87. Through the Figure 13, β can be roughly determined as 30˚. The other parameters refer to Table 1 and Table 2. According to the formula in the literature [12] , it can be calculated that the landslide thrust is 195.7 kN/m. The load distribution is taken as parabolic distribution and the parabolic

vertex load is adopted as the value we just calculate out. After that, we use the same simulation method as Section 2.1 to calculate. Results display that the maximum Von Mises stress of pipeline reach to 566 MPa, which exceeds the pipe material yield strength more than 2%, and the maximum displacement of pipeline is 0.5212 m. Due to these result values related to the pipeline are larger than those values when the slope is in limit equilibrium state. So these results

Notes: pipe thickness δ (mm), buried depth of pipeline H (m), the limit length of pipeline safely withstanding transverse landslide L₀ (m).

Notes: pipe thickness δ (mm), buried depth of pipeline H (m), the limit length of pipeline safely withstanding transverse landslide L₀ (m).

can indirectly indicate the feasibility and correctness of the modeling in Section 2.1.

Similarly, the limit length of pipeline safely withstanding transverse landslide can be calculated out, when the X80 pipe (D = 1.016 m, δ = 18.2 mm) is buried in different depth. The results are shown in Table 5.