Finite element analysis of wrinkling of buried pressure pipeline under strike-slip fault

Earthquakes may constitute a threat for the structural integrity of buried pipelines [1], such as fault movements, landslides, surface collapse and debris flow. Evaluation of the response of buried pressure pipeline crossing the faults is among their top seismic design priorities [2]. This is because the axial and bending strains induced to the pipeline by fault may become fairly large and lead to rupture, either due to tension or due to buckling. Strike-slip fault is one of the most dangerous earthquake disaster. Buckling and fracture may appear on the buried pipeline under strike-slip fault. It may lead to the leakage of gas or liquid, and then causes explosions, pollution layer and other accidents. The buckling behavior of buried steel pipelines subjected to excessive ground deformation has received significant attention in the pipeline community in the recent year. Vazouras [1, 3] studied the mechanical behavior of buried pipelines crossing active strike-slip faults by finite element method. Karamitros [2] presented an analytical methodology using a combination of beam-on-elastic-foundation and the beam theory. Lillig deemed that the properties of the surrounding soil may have a strong influence on pipeline response, and that soil-pipe interaction should be taken into account [4]. Wang [5] analyzed the strain of buried pipelines under strike-slip faults. Duan [6] presented a design method of subsea pipelines against earthquake fault movement. Sim [7] examined the combined effect of dynamic vibration and shear deformation of the surrounding soil on buried pipelines crossing a vertical fault, which could be used for intelligent design of future pipe networks. But these researches were presented not considering the buckling modes of pipeline. Zhang [8] studied bulking behavior of buried pipeline crossing the thrust fault by finite element method. But the internal pressure was not considered. Especially the pressure pipeline, its buckling modes are different with the non-pressure pipeline. There are three types for the fault, which is the normal fault, thrust fault and strike-slip fault. Compressive stress may cause wrinkling of the buried pipeline either in the beam mode or in the shell mode [9]. Thus, the various simplified methods earlier prove to be inadequate for the analysis of pipeline crossing strike-slip fault. In this paper, wrinkling of buried pipeline crossing the strike-slip fault was investigated by finite element method, considering the soil-pipeline interaction. Effects of fault displacement, internal pressure radius-thickness ratio and fault dip angle on longitudinal strain of buried steel pipeline were discussed. The results can provide pipeline laying, protective engineering and safety assessment with a reference basis.


Introduction
Earthquakes may constitute a threat for the structural integrity of buried pipelines [1], such as fault movements, landslides, surface collapse and debris flow.Evaluation of the response of buried pressure pipeline crossing the faults is among their top seismic design priorities [2].This is because the axial and bending strains induced to the pipeline by fault may become fairly large and lead to rupture, either due to tension or due to buckling.Strike-slip fault is one of the most dangerous earthquake disaster.Buckling and fracture may appear on the buried pipeline under strike-slip fault.It may lead to the leakage of gas or liquid, and then causes explosions, pollution layer and other accidents.The buckling behavior of buried steel pipelines subjected to excessive ground deformation has received significant attention in the pipeline community in the recent year.Vazouras [1,3] studied the mechanical behavior of buried pipelines crossing active strike-slip faults by finite element method.Karamitros [2] presented an analytical methodology using a combination of beam-on-elastic-foundation and the beam theory.Lillig deemed that the properties of the surrounding soil may have a strong influence on pipeline response, and that soil-pipe interaction should be taken into account [4].Wang [5] analyzed the strain of buried pipelines under strike-slip faults.Duan [6] presented a design method of subsea pipelines against earthquake fault movement.Sim [7] examined the combined effect of dynamic vibration and shear deformation of the surrounding soil on buried pipelines crossing a vertical fault, which could be used for intelligent design of future pipe networks.But these researches were presented not considering the buckling modes of pipeline.Zhang [8] studied bulking behavior of buried pipeline crossing the thrust fault by finite element method.But the internal pressure was not considered.Especially the pressure pipeline, its buckling modes are different with the non-pressure pipeline.
There are three types for the fault, which is the normal fault, thrust fault and strike-slip fault.Compressive stress may cause wrinkling of the buried pipeline either in the beam mode or in the shell mode [9].Thus, the various simplified methods earlier prove to be inadequate for the analysis of pipeline crossing strike-slip fault.In this paper, wrinkling of buried pipeline crossing the strike-slip fault was investigated by finite element method, considering the soil-pipeline interaction.Effects of fault displacement, internal pressure radius-thickness ratio and fault dip angle on longitudinal strain of buried steel pipeline were discussed.The results can provide pipeline laying, protective engineering and safety assessment with a reference basis.

Methods and materials
Bending deformation of straight pipeline appears under the bending moment caused by the strike-slip fault.Elastic buckling, elastic-plastic buckling and buckling of straight pipeline may occur under external loads.Many formulas of estimating buckling stress and moment of the cylinder under bending have been explained, but not considering the internal pressure.One of the earliest efforts in nonlinear structural analysis was performed by Brazier [10].He found that this category of cylinders collapsed when the radial inward deflection reached 1/9 of the cylinder diameter.Timoshenko and Gere [11] stared that the maximum compressive stress at the critical buckling moment is about 30% higher than that obtained value.
Those theories were expressed not considering the internal pressure.Based on the plastic theory, the numerical solutions for the stress and strain components are obtained for a typical pressure pipeline by Hu [12].Strain and stress distribution of the internal pressure pipeline can be obtained by formulas.But he didn't research the pipeline buckling problem.The bending deformation of buried pipeline is a nonlinear problem.The bending moment is not uniform along the axial direction of the pipeline.Soil-pipeline interaction is an important factor for the buckling behavior of buried pipeline.In addition, pipeline is a thin shell structure, when the large deformation appears on the cross section of pipeline, superposition principle can't be used for the interaction of axial strain and bending strain.There may be residual stress and stress concentration for the pipeline, therefore, it is difficult to solve the response of buried pipeline by the analytic method, and the finite element method is more suitable.
Schematic diagram of deformed pipeline crossing strike-slip fault area can be seen in Fig. 1.When the fault displacement is small, "snake" behavior of buried pipeline appears.But with the increasing of fault displacement, the buried pipeline may be cut or broken.
The structural response of steel pipeline crossing the strike-slip fault is examined numerically, using the general purpose finite element program ABAQUS.The nonlinear material behavior of the steel pipeline and surrounding soil, the interaction between the soil and steel pipeline, as well as the distortion of the pipeline cross-section and the deformation of the surrounding soil are modeled in a rigorous manner, so that the pipeline performance criteria are evaluated with a high-level accuracy.Four-node reduced-integration shell elements (type S4R) are employed to model the cylindrical pipeline segment, and eight-node reduced-integration elements (C3D8R) are used to simulate the surrounding soil.Buried depth is chosen equal to about 2 pipeline diameter, which is in accordance with pipeline engineering practice [13].The soil length in the x direction is equal to at least 60 pipeline diameters, while dimensions in directions y, z equal to 7.5 and 10 times the pipeline diameter respectively.A total of 48 shell elements around the cylinder circumference in this central part have been found to be adequate to achieve convergence of the solution, whereas the size of the shell elements in the longitudinal direction has been chosen equal to 1/15 of the pipeline outer diameter D.

Fig. 2 Finite element model
The fault plane divides the soil in two blocks of equal size (Fig. 1).The analysis is conducted in two steps as follows, gravity loading is applied to the whole model and internal pressure is loaded on the inner wall of the pipeline firstly.Subsequently fault displacement is imposed.The nodes on the bottom boundary planes of the first block (soil nodes) remain fixed in the y direction.A uniform z direction displacement owing to the fault is imposed at side nodes of the second block.In addition, all nodes on the end boundary plane of both blocks are fixed with respect to x direction.
A large-strain von Mises plasticity model with isotropic hardening is employed for the steel pipeline material.The mechanical behavior of soil material is described through an elastic-perfectly plastic Mohr-Coulomb constitutive model, characterized by the cohesion c, the friction angle , the elastic modulus E, and Poisson's ratio .The dilation angle is assumed equal to zero for cases considered in this paper.The interface between the outer surface of the pipeline and the surrounding soil is simulated with a contact algorithm, which allows separation of the pipeline and soil, and accounts for interface friction, through an appropriate friction coefficient µ.In the majority of results reported in the study, µ is considered equal to 0.30.
Taking loess for example, it has a cohesion c = 24.6 kPa, friction angle  = 11.7°[8], Young's modulus E = 33 MPa, Poisson's ratio  = 0.44, density ρ = = 1400 kg/m 3 .Numerical results are obtained for X80 steel pipelines.The pipeline diameter is 0.9144 m (36 in), which is a typical size for oil and gas transmission pipeline.The pipeline wall thickness t is considered equal to 8 mm.X80 is a typical steel material for oil and gas pipeline applications, with a nominal stress-strain curve shown in Fig.
3 [3].The yield stress σ y of X80 is 596 MPa.Young's modulus of steel material equal to 210 GPa, Poisson's ratio is 0.30, density is 7800 kg/m 3 [14,15].Considering a safety factor equal to 0.72, and the maximum operating pressure P max of this pipeline, given by P max =0.722σ y t/D.The bending moment increases with the increasing of the fault displacement.For pipeline wrinkling of B part, strain of the upper part is compression strain, while the lower part is tension strain.But for the wrinkling of A part, strain of the lower part is compression strain, while the upper part is tension strain.The amplitudes of the wrinkles increase with the increasing of the fault displacement.Wrinkles may reduce the strength of steel pipeline and increase the difficulty of pigging.If the plastic strain is bigger than the rupture strain, the leakage will happen.Deflection curves of the buried pipeline under different fault displacements are shown in Fig. 6.When the fault displacement is small, the deflection curve is smooth S-shape.But with the increasing of the fault displacement, the deflection curve is non-smooth Z-shape.There are two inflection points in the deflection curve.The local wrinkling locations are the two inflection points.So, wrinkling is more serious with the increasing of the strike-slip fault displacement.The lengths of A and B part increase gradually also.Therefore, the simplified calculation model [1] was put forward by Vazouras is not suitable for the buckling analysis under strike-slip fault displacement.
The variations of longitudinal strain along the two outer generators of the pipeline in B part are shown in Fig. 7.At the compression side (Fig. 7, a), the outset of local wrinkling is considered at the stage where outward displacement of the pipeline wall starts at the area of maximum compression.At that stage, bending strains due to pipeline wall wrinkle develop, associated with significant tensile strains at the "ridge" of the wrinkling.So that the longitudinal compressive strains at this location at the outer surface start decreasing, forming a short wave at this location [1].At the ridge of the wave, the longitudinal strain is the negative maximum value.While at the valley, the longitudinal strain is the positive maximum value.The wave amplitude increases with the increasing of the strike-slip fault displacement.Meanwhile, the tensile strain of valley bottom and the compression strain of the crest increase with the increasing of the fault displacement.At the tension side (Fig. 7, b), the longitudinal strain increases with the increasing of the strike-slip fault displacement.
When the fault displacement is more than 1.8 m, change of the longitudinal strain on both sides of the crest is small.

Internal pressure effect
When the diameter-thick ratio D/t = 114, strike-slip fault displacement u = 3.0 m, dip angle β=45°, wrinkling modes of the buried pipeline under different pipeline internal pressures are shown in Fig. 8.The wrinkling morphologies of A and B part pipeline are different under different internal pressures.From the local amplification figure of A and B wrinkling, the wrinkling mode of the pipeline presents three states under different internal pressures.The first one is collapse buckling when the internal pressure is zone or small.The second one is the critical state between collapse and wrinkling when the internal pressure is less than 0.6P max .The third one is wrinkling when the internal pressure is more than 0.8P max .In this state, only one wrinkle on A part, but more than two wrinkles on B part.The distance between A and B wrinkling location is different under different internal pressures.It decreases with the increasing of internal pressure.The internal pressure can enhance the stiffness of buried steel pipeline to resistance to bending moment caused by strike-slip fault displacement.Therefore, the internal pressure has a great effect on the buckling mode of the buried pipeline under fault displacement.

Fig. 8 Wrinkling modes of pipeline under different internal pressures
The variations of longitudinal strain along the two outer generators of the buried steel pipeline in B part are shown in Fig. 9.At the compression side (Fig. 9, a), when P = 0, there is only one crest, and the tensile strain is big and the compression strain is small.When P ≥ 0.2P max , the tensile strain is smaller than the compression strain.When P = 0.2P max , the tensile strain is also big, which illustrates that the buckling pattern is collapse.The strain curves of 0.4P max and 0.6P max are more similar, the compression strain is more than tensile strain.It shows that the wrinkling occurs but the collapse does not disappear.There are two peaks of compression strain when P = 0.8P max .It shows that there are two wrinkles appear on the compression side of the pipeline.When P = P max , there are three peaks of compression strain.At the tensile side (Fig. 9, b), the maximum longitudinal strain increases with the increasing of internal pressure.The longitudinal strain is very small when the internal pressure is zone.When P = 0.2P max , the curve shape is a bimodal curve.When P > 0.2P max , , and the bimodal curve becomes to unimodal curve.The maximum tensile strain is smaller than the maximum compression strain.a b Fig. 9 Longitudinal strain distribution of wrinkling in B part under different internal pressures a) Longitudinal strain at the compression side; b) Longitudinal strain at the tension side Table 1 shows the maximum longitudinal strain under different pressures and fault displacements.Longitudinal strain of buried pipeline with bigger internal pressure is greater under the same fault displacement.For high pressure buried pipeline, change rate of longitudinal strain is bigger along with fault displacement.When strike-slip fault displacement u = 3 m, the internal pressure is P max , buckling modes of the buried oil-gas pipeline under different diameter-thick ratios are shown in Fig. 10.With the decreasing of the diameter-thick ratio, the bending curve of the pipeline is more smooth.The greater diameter-thick ratio can enhance the ability to resistance to bending moment.The lengths of A and B part increase with the increasing of the diameter-thick ratio.When the diameter-thick ratio is small, there are more wrinkles in A and B part, but the wrinkling is not serious.The amplitude of the wrinkle decreases with the decreasing of the diameter-thick ratio.Therefore, buried steel pipeline is more safety with a small diameter-thick ratio.For the dangerous area, transportation safety can be ensured by increasing the wall thickness of the buried steel pipeline.Fig. 10 Wrinkling modes of pipeline under different diameter-thick ratios Deflection curves of the buried pipeline under different diameter-thick ratios can be seen in Fig. 11.Under the same fault displacement, the deflection curve is more smooth with the increasing of the diameter-thick ratio.It means that the buried steel pipeline with a smaller diameter-thick ratio could resist the deformation caused by strike-slip fault displacement.The curve shape of the buried pipeline changes from S-shape to Z-shape with the increasing of the diameter-thick ratio.The inflection points of the bending curve mean the wrinkling locations.The center point of the pipeline (it in the fault plane) is not changed along with the diameter-thick ratio.The curves Fig. 11 Deflection curves of buried pipeline under different diameter-thick ratios are axially symmetric distribution around the center point.The variations of longitudinal strain along the two outer generators of the pipeline in part B under different diameter-thick ratios are shown in Fig. 12.At the compression side (Fig. 12, a), the wave of the strain curve becomes smooth gradually with the decreasing of the diameter-thick ratio.When D/t ≤ 59, the longitudinal strain is very small.At the tensile side (Fig. 12, b), the axial tensile strain decreases with the decreasing of the diameter-thick ratio.It illustrates that the probability of pipeline wrinkling is bigger with a lower diameter-thick ratio.Table 2 shows the maximum longitudinal strain under different diameter-thick ratios and fault displacements.Longitudinal strain of buried pipeline can be reduced by decreasing diameter-thick ratio.Thick wall pipelines have superior ability to resistance to deformation under bending moment.Therefore, thick wall pipelines should be used in fault area.When the diameter-thick ratio D/t = 114, strike-slip fault displacement u = 3 m, internal pressure is P max , dip angle β = 0°, 15°, 30° and 45°, the variations of longitudinal strain along the two outer generators of the pipeline in B part are shown in Fig. 13.At the compression side (Fig. 13, a), wrinkling modes of β = 0° and 15° are similar, longitudinal compression strain of the first wrinkle is the biggest, the second one is smaller, and the third a b Fig. 13 Longitudinal strain distribution of wrinkling in B part under different dip angles: a) Longitudinal strain at the compression side; b) Longitudinal strain at the tension side one is the smallest.Longitudinal compression strain of β = 15° is bigger than β = 0°.Wrinkling modes of β = 30° and 45° are similar, longitudinal compression strain of the second wrinkle is the biggest, the others are small.The maximum longitudinal compression strain of β = 45° is bigger than β = 30°.Therefore, there are two wrinkling modes of the B part pipeline when β=0°~45°.At the tensile side (Fig. 13, b), the maximum longitudinal tensile strain of β=15° is bigger than β = 0°, and the maximum longitudinal tensile strain of β = 45° is bigger than β = 30°.Table 3 shows the maximum longitudinal strain under different dip angles and fault displacements.Effect of dip angle on the longitudinal strain of buried pipeline is smaller.Under bigger fault displacement, buried pipeline is prone to be cut by the fault plane.

Conclusions
Finite element analysis of wrinkling of the buried pressure pipeline under strike-slip fault displacement caused by earthquake was investigated in this paper.Effects of strike-slip fault displacement, internal pressure, diameter-thick ratio and dip angle on the wrinkling modes and longitudinal strain of the buried steel pipeline were discussed.That led to the following conclusions: 1.There are two local wrinkling parts of the buried pipeline under strike-slip fault displacement.The wrinkling is more serious with the increasing of the strike-slip fault displacement.There is only one wrinkle on the A part, but more than two wrinkles on B part.The stain wave amplitude of B part increases with the increasing of the strike-slip fault displacement.
2. Buckling mode of buried steel pipeline presents three states under different internal pressure.The first one is collapse buckling when the internal pressure is zone or small.The second one is the critical state between collapse and wrinkling when the internal pressure is less than 0.6P max .The third one is wrinkling when the internal pressure is more than 0.8P max .
3. The greater diameter-thick ratio can enhance the ability to resistance to bending moment.The length of A and B part increases with the increasing of the diameter-thick ratio.The amplitude of the wrinkle decreases with the decreasing of the diameter-thick ratio, and the curve shape of the pipeline changes from S-shape to Z-shape.
4. At the compression side of B part, wrinkling modes of β = 0° and 15° are similar, axial compression strain of the first wrinkle is the biggest, the second one is smaller, and the third one is the smallest.Wrinkling modes of β = 30° and 45° are similar, axial compression strain of the second wrinkle is the biggest, the others are small.

Fig. 1
Fig. 1 Schematic diagram of deformed pipeline crossing fault area In this paper, define fault dip angle β = 45°, pipeline crossing angle  = 90°.Fig. 2 shows the finite element models of the buried pipeline and fault layer.The pipeline is embedded in an elongated soil prism along the x axis.Four-node reduced-integration shell elements (type S4R) are employed to model the cylindrical pipeline segment, and eight-node reduced-integration elements (C3D8R) are used to simulate the surrounding soil.Buried depth is chosen equal to about 2 pipeline diameter, which is in accordance with pipeline engineering practice[13].The soil length in the x direction is equal to at least 60 pipeline diameters, while dimensions in directions y, z equal to 7.5 and 10 times the pipeline diameter respectively.A total of 48 shell elements around the cylinder circumference in this central part have been found to be adequate to achieve convergence of the solution, whereas the size of the shell elements in the longitudinal direction has been chosen equal to 1/15 of the pipeline outer diameter D.

Fig. 3
Fig. 3 Strain-stress curve of X80 3. Results and discussions 3.1.Strike-slip fault effect When the diameter-thick ratio D/t = 114, pipeline pressure P max = 7.5 MPa, dip angle β = 45°, wrinkling modes of the buried oil-gas pipeline under different strike-slip fault displacements are shown in Fig. 4.There are two local wrinkling parts of the buried pipeline under strike-slip fault displacement.With the increasing of the fault displacement, wrinkling appears on B part of the pipeline firstly.Then it appears on A part.There is only one wrinkle on the A part, but three wrinkles on B part.The bending moment increases with the increasing of the fault displacement.For pipeline wrinkling of B part, strain of the upper part is compression strain, while the lower part is tension strain.But for the wrinkling of A part, strain of the lower part is compression strain, while the upper part is tension strain.The amplitudes of the wrinkles increase with the increasing of the fault displacement.Wrinkles may reduce the strength of steel pipeline and increase the difficulty of pigging.If the plastic strain is bigger than the rupture strain, the leakage will happen.

Fig. 4
Fig. 4 Wrinkling modes of pipeline under different fault displacements Fig. 5 shows the von Mises stress of the buried steel pipeline under different fault displacements.When u = 0.6 m, the maximum von Mises stress appears on the upper part in part B of the steel pipeline.When u = 1.2 m, there are two high stress area, they are on the upper part in part B and the lower part in part A. The maximum stress increases with the increasing of strike-slip fault displacement.When u ≥ 1.8 m, the von Mises stress distribution of the buried steel pipeline changes little.The high stress areas decrease with the increasing of strike-slip fault displacement.The maximum stresses are mainly on the wrinkling part of the steel pipeline.Therefore, failure parts of the buried steel pipeline are mainly concentrated in the two wrinkling parts after buckling appears.

Fig. 5
Fig. 5 Von Mises stress of pipeline under different fault displacements

Fig. 6
Fig. 6 Deflection curves of buried pipeline under different fault displacements Longitudinal strain distribution of wrinkling in B part under different diameter-thick ratios: a) Longitudinal strain at the compression side; b) Longitudinal strain at the tension side

Table 1
The maximum longitudinal strain under different pressures

Table 3
The maximum longitudinal strain under different dip angles