# Predicting the Instability Trajectory of an Obliquely Loaded Pipeline on a Clayey Seabed

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Modelling

#### 2.1. Finite Element Mesh and Boundary Conditions

_{u}(including the submerged pipe weight) and the corresponding pipe movement, respectively. The angle of the slip zone boundary (∠BOA), the “zero” shear stress point (∠BOC), the load direction (∠BOD), the pipe embedment (∠BOE) and the movement direction (∠BOF) are termed as ε

_{s}, ε

_{f}, ε

_{F}, θ

_{0}and ε

_{u}, respectively. The counterclockwise direction from OB is defined positive.

_{u}. The value of α ranges from 0 for a smooth interface to 1.0 for a fully rough interface. In design, it is common to assume that a pipe–soil interface extending to the free surface cannot sustain tension, because a crack may open [13]. In the present simulation, the pipe–soil interface can sustain shear stress αs

_{u,}but cannot sustain tension. For the aluminum pipe commonly used in pipe–soil interaction tests, α is approximately in the range of 0.30 to 0.36 [33,34]. A kinematic coupling constraint was imposed between the nodes of model pipe and the reference point at the pipe center. The translational displacement of the model pipe was applied on the reference point, then transferred to the pipe section by the coupling constraint.

#### 2.2. Constitutive Models

_{u}.

_{ij}represents the normal and shearing stress in the three-dimensional Cartesian coordinate system, and ε

_{ij}represents the normal and shearing strain. J is the deviatoric stress and Θ is Lode’s angle. σ′

_{1}, σ′

_{2}, and σ’

_{3}are the first, second and the third principal stress, respectively. The elastic behavior is described by the Poisson’s ratio ν

_{s}= 0.49, and the Young’s modulus of the soil E

_{s}= 50 MPa. As E

_{s}barely affects the ultimate bearing capacity of the pipe [35,36], the present relatively large value of E

_{s}= 50 MPa was adopted to ensure that the soil’s elastic deformation would not change the initial embedment of the model pipe.

#### 2.3. Properties of the Pipe and the Clayey Soil

_{u}were the independent variables. A wished-in-place model pipe was set on the flat seabed with initial embedment no more than 0.5D. The model pipe was then obliquely loaded to move relative to the seabed in a specific direction, which is characterized by ε

_{u}. The value of ε

_{u}was zero for the vertical penetration (see Figure 1). The model pipe displacement, soil resistance, normal and shear stresses on the pipe–soil interface, equivalent plastic strain (PEEQ) were recorded synchronously. The numerical simulation was terminated after the soil resistance reached its residual value.

## 3. Results and Discussion

#### 3.1. Failure Mechanism

_{u}= 0° (vertical), 30° (oblique), 60° (oblique), and 90° (horizontal). The soil resistance gradually approaches its ultimate value with the increase in pipe displacement. The maximum value of the ultimate bearing capacity was achieved for the case of vertical penetration. As the load direction shifts from vertical (ε

_{u}= 0°) to horizontal (ε

_{u}= 90°), the ultimate soil resistance decreases significantly.

_{u}= 30°, see Figure 3c), the slip mechanism first emerges on the front side and then a plastic zone is observed on the back side as the soil resistance approaches its residual value. The area of plastic zone on the back side decreases rapidly as ε

_{u}increases. The position of the slip zone boundary within the soil can be characterized by ε

_{s}(see Figure 1). The relationship between ε

_{s}and the movement angle (ε

_{u}) is presented in Figure 4. It is indicated that for ε

_{u}> 45°, the calculated values of ε

_{s}match well with the prediction by Equation (5):

#### 3.2. Ultimate Bearing Capacity for Vertical and Horizontal Instability

_{u}= 0°) and the horizontal swipe (ε

_{u}= 90°) are shown in Figure 5 and Figure 6, respectively. The vertical component of the ultimate bearing capacity is termed as V, while the horizontal component is denoted by H.

_{u}, the shearing and normal stress on the pipe–soil interface can be calculated as follows:

_{A}is the mean stress on the seabed surface. θ is the position angle of the point on the pipe–soil interface (θ = 0 for the bottom point). Assuming that the shear stress is uniformly distributed on the pipe–soil interface, the normalized vertical bearing capacity V/s

_{u}D for e/D < 0.5 can be predicted by:

_{u}= π/2. The absolute value of the shear stress reaches its maximum αc (= 3.20 kPa) on the edge of the pipe–soil interface, while the directions of the local shear stress vary along the interface. The numerical results indicate that, for ε

_{u}< 45°, the shear stress along the pipe–soil interface has different directions on the left and right side of the pipe’s geometric center. With the increase in ε

_{u}, the plastic zone within the soil reduces (see Figure 3), while the direction of the shear stress on the interface tends to be identical (see Figure 8; note: ε

_{f}represents the position where the shear stress is zero). Referring to Figure 8, the correlation of ε

_{f}with ε

_{u}can be represented by Equation (9):

_{f}and ε

_{u}(see Figure 8), for the horizontal swipe case, the magnitude of the contact shear stress τ on the pipe–soil interface can be reasonably assumed to be αc with the direction of τ reversing at the position θ = π/4. Note that the pipe and its underlying soil surface disconnect with each other at θ = 0. By integrating σ and τ using Equations (6) and (7) along the pipe–soil interface, the dimensionless horizontal and vertical components of the ultimate bearing capacity for the horizontal swipe tests can be calculated by Equations (10) and (11), respectively:

#### 3.3. Failure Envelopes

_{u}approaches its maximum (θ

_{0}+ π/2), negative values of V were observed in the present numerical simulations (see Figure 9), especially for the rough pipe–soil interface and the large initial embedment of the pipe. As shown in Figure 9a–c, with the increase in the pipe embedment (e/D) from 0.1 to 0.5, the bearing capacity for the lateral instability (H/s

_{u}D) increases more significantly than that for the vertical penetration (V/s

_{u}D). For the fully rough pipe–soil interface (α = 1.0), the values of V

_{m}/H

_{m}are 4.96 and 2.85 for the case e/D = 0.1 and 0.5, respectively. V

_{m}is the maximum vertical ultimate bearing capacity and H

_{m}is the maximum horizontal bearing capacity of the failure envelope. For the smooth pipe (α = 0), the values of V

_{m}/H

_{m}are 6.22 and 2.75 for the case e/D = 0.1 and 0.5, respectively. As ε

_{u}approaches its maximum value θ

_{0}+ π/2, the value of ε

_{s}increases and the plastic failure zone within the underlying soils becomes smaller.

_{m}/s

_{u}D, see Figure 9), the length of the long axis (V

_{m}/s

_{u}D) and the equation of the envelope can be expressed bys Equations (14)–(16), respectively:

_{s}), the critical horizontal breakout load F

_{br}can be further derived from Equation (16):

_{m}and V

_{m}are in correlation with the diameter (D) and the embedment (e) of the pipe, the undrained shear strength (s

_{u}) of the soil and the interface friction ratio (α) of the pipesoil interface (see Equations (14) and (15)). The overpenetration ratio can be affected by the laying process and the spanning of a pipeline caused by scour or upheaval buckling [42].

#### 3.4. Trajectory of Pipe Instability: Critical Submerged Weight of the Pipe

_{F}) with the pipe movement angle (ε

_{u}) for various values of α are shown in Figure 10. It is indicated that the ultimate load angle ε

_{F}is in the positive correlation with the movement angle ε

_{u}, the dimensionless embedment e/D and the roughness coefficient of the pipe–soil interface α. An empirical equation for such correlation is established:

_{F}and ε

_{u}are both in radians. The corresponding curves predicted with Equation (19) are compared with the numerical results in Figure 10. A good consistency can be observed.

_{u}is equal to π/2 with the pipe losing lateral stability. According to Equation (19), the load direction of the pipe under the ultimate loading can be expressed as:

_{F-pre}) match well with the numerical results (ε

_{F-num}).

_{u}), respectively. Note that V includes the submerged weight per unit length of the pipe (W

_{s}) and the vertical component of the external load (V

_{ex}), i.e., V = W

_{s}+ V

_{ex}. The comparison between the predicted values of the non-dimensional ultimate load (F

_{u-pre}/s

_{u}D) and numerical results (F

_{u-num}/s

_{u}D) is illustrated in Figure 12, indicating a good consistency.

_{cr}= W

_{scr}/s

_{u}D) can be derived:

_{m}/W

_{scr}represents the critical overpenetration ratio R

_{cr}, i.e.,

_{cr}= V

_{m}/W

_{scr}

_{m}, V

_{m}and ε

_{F}depend on the dimensionless pipe embedment (e/D) and the interface friction ratio of the pipe–soil interface (α) (see Equations (14), (15) and (20)). Note that V

_{ex}= 0 if the external load is in the horizontal direction. The value of ε

_{F}can be obtained by Equation (20). The pipe has the tendency to rise during its lateral instability, when its submerged weight per unit length is less than the critical value W

_{scr}. Otherwise, the pipe would move downwards when it is heavier than the critical submerged weight.

_{0}/D), the submerged weight of the pipe (W

_{s}), the undrained strength of the clayey soil (s

_{u}), the interface friction ratio of the pipe–soil interface (α), and the external load (F). For a partially embedded pipe under the horizontal external loading, the movement trajectory during lateral instability can be calculated using an iterative algorithm based on Equations (19) and (23). The developments of the embedment ratio (e

_{x}/D) with the horizontal component of pipe displacement (u

_{x}) for e

_{0}/D = 0.2 and various values of R at α = 0, 0.5, and 1.0 are illustrated in Figure 13a–c, respectively. Note that the overpenetration ratio R is in negative correlation with the submerged pipe weight W

_{s}(see Equation (18)).

_{cr}can be solved with Equation (25). For the case R > R

_{cr}, the pipe rises to a constant embedment during its lateral instability, while the pipe moves downward for R < R

_{cr}. The values of R

_{cr}are 2.00, 2.41 and 3.54 for the cases of α = 0, 0.5, 1.0, respectively. That is, R

_{cr}is in the positive correlation with α.

_{ex}= 0), the variations of G

_{cr}with e/D for various values of α are shown in Figure 14. The effect of the pipe embedment ratio to the dimensionless critical submerged weight of the pipe decreases when the values of e/D become larger. The pipe with a rougher surface generally tends to plough into the soil (i.e., the corresponding values of G

_{cr}become smaller), while the smoother pipe is more likely to move upwards.

## 4. Conclusions

- (1)
- As the pipe movement angle ε
_{u}increases from 0° (vertical) to 90° (horizontal), the spatial range of the slip zone in front of the partially embedded pipe decreases gradually, and the corresponding ultimate soil resistance decreases significantly. The angle of slip zone boundary ε_{s}and the pipe movement angle ε_{u}can be linearly correlated (see Equation (5)). It was found that the direction of shear stress varies along the pipe–soil interface. The angle of the “zero” shear stress point ε_{f}generally keeps constant for ε_{u}< π/4 and increases linearly with increasing ε_{u}for ε_{u}> π/4. - (2)
- Numerical results indicate that the shape of bearing capacity envelopes resembles an ellipse, which can be empirically described with Equation (16). A slip-line field solution of the bearing capacity for the horizontal swipe was further derived. The slip-line field solutions agree well with the present numerical results, indicating the prospect of the derived slip-line field solution for predicting the horizontal swipe failure.
- (3)
- An empirical expression of the load angle ε
_{F}in correlation with ε_{u}, the embedment ratio e/D and the interface friction ratio α, was proposed to characterize the flow rule of a force-resultant plasticity model for predicting the pipe behavior. The trajectories of the pipes with different submerged weight under the horizontal loading were obtained. Based on the established bearing capacity envelope and the flow rule, an analytical solution of the critical submerged weight of the pipe was finally obtained for distinguishing the “light” and the “heavy” pipes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$D$ | Outer diameter of the pipe |

$e$ | Pipe embedment |

${e}_{\mathrm{x}}$ | Pipe embedment during lateral movement |

${E}_{\mathrm{p}}$ | Young’s modulus of the pipe |

${E}_{\mathrm{s}}$ | Young’s modulus of the soil |

$f$ | Shearing stress at the pipe–soil interface |

$F$ | Total load upon the pipe |

${F}_{\mathrm{br}}$ | Horizontal load inducing pipe breakout |

${F}_{\mathrm{u}}$ | Total ultimate load |

${F}_{\mathrm{u}\text{-}\mathrm{num}}$ | Numerical result of the ultimate load |

${F}_{\mathrm{u}\text{-}\mathrm{pre}}$ | Predicted value of the ultimate load |

${G}_{\mathrm{cr}}$ | Critical dimensionless submerged pipe weight |

$h$ | Depth of the soil model |

$H$ | Horizontal component of the ultimate bearing capacity |

${H}_{\mathrm{m}}$ | Maximum horizontal bearing capacity of the failure envelope |

$J$ | Deviatoric stress |

$l$ | Width of the soil model |

$R$ | Overpenetration ratio |

${R}_{\mathrm{cr}}$ | Critical overpenetration ratio |

${s}_{\mathrm{u}}$ | Undrained strength of the clayey soil |

$u$ | Displacement of the pipe |

${u}_{\mathrm{x}}$ | Horizontal component of the pipe displacement |

$V$ | Vertical component of the ultimate bearing capacity |

${V}_{\mathrm{ex}}$ | Vertical component of the external ultimate load |

${V}_{\mathrm{m}}$ | Maximum vertical component of the bearing capacity of the failure envelope |

${W}_{\mathrm{s}}$ | Submerged pipe weight per unit length |

${W}_{\mathrm{scr}}$ | Critical submerged weight of the pipe per unit length |

$\alpha $ | Interface friction ratio |

$\beta $ | Parameter in Equation (21) |

${\sigma}_{\mathrm{ij}}$ | Normal and shearing stress |

${{\sigma}^{\prime}}_{1}~{{\sigma}^{\prime}}_{3}$ | Principal stresses of the soil |

$\Delta $ | The value equal to $\mathrm{arcsin}\alpha $ |

${\epsilon}_{\mathrm{ij}}$ | Normal and shearing strain |

${\epsilon}_{\mathrm{f}}$ | Angle of “zero” shear stress point |

${\epsilon}_{\mathrm{F}}$ | Angle of the load direction |

${\epsilon}_{\mathrm{s}}$ | Angle of the slip zone boundary |

${\epsilon}_{\mathrm{u}}$ | Movement angle |

${\epsilon}_{\mathrm{ucr}}$ | Critical movement angle |

${v}_{\mathrm{p}}$ | Poisson’s ratio of the pipe |

${v}_{\mathrm{s}}$ | Poisson’s ratio of the soil |

$\theta $ | Position angle of an arbitrary point on the pipe–soil interface |

${\theta}_{0}$ | Half of embedment angle |

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**Figure 1.**Schematic diagram of the pipe–soil interaction system with the pipe moving in an obliquely downward direction.

**Figure 3.**(

**a**) The non-dimensional soil resistance vs. the non-dimensional pipe displacement for obliquely loaded pipe–soil interactions; the evolutions of equivalent plastic strain zone (α = 1.0, e/D = 0.4): (

**b**) the pipe movement angle ε

_{u}= 0° (vertical penetration); (

**c**) ε

_{u}= 30°; (

**d**) ε

_{u}= 60°; (

**e**) ε

_{u}= 90° (horizontal swipe).

**Figure 4.**The relationship between the position of the slip zone boundary (ε

_{s}) and the pipe movement angle (ε

_{u}).

**Figure 5.**Variations of the normalized vertical bearing capacity (V/s

_{u}D) with the relative embedment (e/D) for various values of pipe roughness (α) in the vertical penetration.

**Figure 6.**Variations of (

**a**) the horizontal component and (

**b**) the vertical component of the normalized ultimate bearing capacity with the relative embedment (e/D) for various values of the pipe roughness (α) in the horizontal swipe.

**Figure 7.**The distribution of shear stress along the pipe–soil interface under the ultimate state for the movement angle ε

_{u}= π/2 (e/D = 0.4, α = 1.0).

**Figure 8.**The correlation between the angle for zero shear stress point (ε

_{f}) and the pipe movement angle (ε

_{u}).

**Figure 9.**The bearing capacity envelopes for the pipe with various embedment ratios (e/D): (

**a**) e/D = 0.5; (

**b**) e/D = 0.3; (

**c**) e/D = 0.1. Note: For the vertical penetration (ε

_{u}= 0) and the horizontal swipe (ε

_{u}= π/2) cases, the moving directions of the pipe are specially marked with pink arrows.

**Figure 10.**The correlation of pipe movement angle (ε

_{u}) with the direction angle of ultimate load (ε

_{F}) for various values of the pipe embedment ratio (e/D): (

**a**) e/D = 0.5; (

**b**) e/D = 0.4; (

**c**) e/D = 0.3; (

**d**) e/D = 0.2; (

**e**) e/D = 0.1.

**Figure 11.**Comparison between the predicted direction angle of the ultimate load (ε

_{F-pre}) and the numerical results (ε

_{F-num}).

**Figure 12.**Comparison between the predicted values of the non-dimensional ultimate load (F

_{u-pre}/s

_{u}D) and the numerical results (F

_{u-num}/s

_{u}D).

**Figure 13.**Developments of the embedment ratio (e

_{x}/D) with the horizontal displacement of the pipe (u

_{x}) for e/D = 0.2 and various values of the overpenetration ratio (R): (

**a**) α = 0; (

**b**) α = 0.5; (

**c**) α = 1.0.

**Figure 14.**Variations of the non-dimensional critical submerged weight of the pipe (G

_{cr}) with the embedment ratio (e/D) for various values of the interface friction factor (α).

Parameters | Symbols | Units | Values |
---|---|---|---|

Outer diameter of the pipe | D | m | 0.46 |

Elastic modulus of the pipe | E_{p} | GPa | 210 |

Poisson’s ratio of the pipe | ν_{p} | -- | 0.19 |

Sumerged weight of the pipe | W_{s} | N/m | Varied in Section 3.4 |

Elastic modulus of the soil | E_{s} | MPa | 50 |

Poisson’s ratio of the soil | ν_{s} | -- | 0.49 |

Undrained shear strength of the soil | s_{u} | kPa | 3.2 |

Pipe–soil interface friction ratio | α | -- | 0, 0.5, 1.0 |

Embedment of the pipe | e | m | 0.1, 0.2, 0.3, 0.4, 0.5 |

Half of embedment angle | θ_{0} | -- | arccos (1 − 2e/D) |

Angle of instability | ε_{u} | -- | 0 ~ (π/2 + θ_{0}) |

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**MDPI and ACS Style**

Wang, N.; Qi, W.; Gao, F.
Predicting the Instability Trajectory of an Obliquely Loaded Pipeline on a Clayey Seabed. *J. Mar. Sci. Eng.* **2022**, *10*, 299.
https://doi.org/10.3390/jmse10020299

**AMA Style**

Wang N, Qi W, Gao F.
Predicting the Instability Trajectory of an Obliquely Loaded Pipeline on a Clayey Seabed. *Journal of Marine Science and Engineering*. 2022; 10(2):299.
https://doi.org/10.3390/jmse10020299

**Chicago/Turabian Style**

Wang, Ning, Wengang Qi, and Fuping Gao.
2022. "Predicting the Instability Trajectory of an Obliquely Loaded Pipeline on a Clayey Seabed" *Journal of Marine Science and Engineering* 10, no. 2: 299.
https://doi.org/10.3390/jmse10020299